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Holomorphic curves in Exploded Torus
Fibrations: Compactness
Brett Parker
November 1, 2018
Abstract
The category of exploded torus fibrations is an extension of the cat-egory of smooth manifolds in which some adiabatic limits look smooth.(For example, the type of limits considered in tropical geometry appearsmooth.) In this paper, we prove a compactness theorem for (pseudo)-holomorphic curves in exploded torus fibrations. In the case of smoothmanifolds, this is just a version of Gromov’s compactness theorem in atopology strong enough for gluing analysis.
Contents
1 Introduction 1
2 Definitions 3
2.1 Tropical semiring and exploded semiring . . . . . . . . . . . . . . 32.2 Exploded T fibrations . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Ck,δ regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Almost complex structures . . . . . . . . . . . . . . . . . . . . . . 212.6 Fiber product, refinements . . . . . . . . . . . . . . . . . . . . . . 242.7 Symplectic structures . . . . . . . . . . . . . . . . . . . . . . . . 282.8 Moduli stack of smooth exploded T curves . . . . . . . . . . . . . 33
3 Estimates 36
4 Compactness 54
1 Introduction
As the subject of this paper is holomorphic curves in a new category, thereader desiring rigorous statements will be disappointed in this introduction,and should wait until terms have been defined.
Holomorphic curves have been a powerful tool in symplectic topology sinceGromov’s original paper on the subject, [5]. The basic idea is that given asymplectic manifold, there is a contractible choice of almost complex structure so
1
that the symplectic form is positive on holomorphic planes. The ‘moduli space’of (pseudo)-holomorphic curves with a choice of one of these almost complexstructures is in some sense ‘smooth’, ‘compact’ and finite dimensional, andtopological invariants of this ‘moduli space’ give invariants of the underlyingsymplectic manifold.
The number of quotation marks required for the above statements is perhapsone indication that the category of smooth manifolds is not the ideal setting forworking with holomorphic curves. In particular, in order for our moduli spaceto be ‘compact’, we have to include connected ‘smooth’ families of holomorphiccurves which exhibit bubbling behavior, and change topology. This certainlydoes not fit the usual definition of a smooth family (although it is more naturalfrom the perspective of algebraic geometry). The topology in which the modulispace is compact is also so unnatural to the smooth category that it is difficultto state simply.
A second reason to leave the smooth category is that holomorphic curves canbe very difficult to find directly there. A common technique used to find holo-morphic curve invariants is to consider a family of almost complex structureswhich degenerates somehow to a limit in which holomorphic curves become sim-pler, and then reconstruct holomorphic curve invariants from limiting informa-tion. In the category of exploded torus fibrations, some of these degenerationscan be viewed as smooth families, and the limiting problem is a problem of thesame type.
One example of such a degeneration is that considered in the algebraic caseby [10]. This is a degeneration represented by a flat family of projective schemesso that the fiber at 0 has two components intersecting transversely over a smoothdivisor, and all other fibers are smooth. This corresponds in the symplecticsetting to the degeneration used to represent a symplectic sum, used in [7]to prove the symplectic sum formula for Gromov-Witten invariants. Similardegenerations have been used to study holomorphic curves in [9] and [2]. Moregenerally, one might consider the case of a flat family in which the central fiberis reduced with simple normal crossing singularities, or the analogous case in thesymplectic setting. This type of degeneration can be represented by a smoothfamily in the category of exploded torus fibrations. In the exploded category,an object B corresponding to this central fiber will have a smooth part ⌈B⌉which is equal to the central fiber, and a tropical part ⌊B⌋ which is a stratifiedintegral affine or ‘tropical’ space which has a vertex for every component ofthe central fiber ⌈B⌉, an edge for every intersection, and an n dimensional facefor each complex codimension n intersection of some number of components.Holomorphic curves in B look like usual holomorphic curves projected to ⌈B⌉,and tropical curves projected to ⌊B⌋.
This tropical part ⌊B⌋ is related to the large complex structure limit studiedin some approaches to mirror symmetry such as found in [6], [4] or [8]. Study-ing the tropical curves in ⌊B⌋ can sometimes give complete information aboutholomorphic curve invariants. One example of this is in [11].
The goal of this paper is to provide the relevant compactness theorems forholomorphic curves in exploded torus fibrations analogous to Gromov’s com-pactness theorem ( stated precisely on page 54). This is one step towards havinga good holomorphic curve theory for exploded torus fibrations. It is anticipatedthat the moduli stack of holomorphic curves in exploded fibrations also admitsa natural type of Kuranishi structure which together with these compactness
2
results will allow the definition of holomorphic curve invariants such as GromovWitten invariants.
2 Definitions
2.1 Tropical semiring and exploded semiring
Functions on exploded fibrations will sometimes take values in the followingsemirings:
The tropical semiring is a semiring tR which is equal to R with ‘multiplica-tion’ being the operation of usual addition and ‘addition’ being the operationof taking a minimum. We will write elements of tR as tx where x ∈ R. Then wecan write the operations as follows
txty := tx+y
tx + ty := tminx,y
t1 can be thought of as something which is infinitesimally small. With this inmind, we have the following order on tR:
tx > ty when x < y
Given a ring R, the exploded semiring RtR consists of elements ctx withc ∈ R and x ∈ R. Addition and multiplication are as follows:
c1txc2t
y = c1c2tx+y
c1tx + c2t
y =
c1tx if x < y
(c1 + c2)tx if x = y
c2ty if x > y
It is easily checked that addition and multiplication are associative and obeythe usual distributive rule. We will mainly be interested in CtR and RtR.
There are semiring homomorphisms
Rι−→ RtR
⌊·⌋−−→ tR
defined byι(c) := ct0
⌊ctx⌋ := tx
The homomorphism ⌊·⌋ is especially important. We shall call ⌊ctx⌋ = tx thetropical part of ctx. There will be an analogous ‘tropical part’ of ‘basic’ explodedfibrations which can be thought of as the large scale.
Define the positive exploded semiring RtR+
to be the sub semiring of RtR
consisting of elements of the form ctx where x ≥ 0. There is a semiring homo-morphism
⌈·⌉ : RtR+ −→ R
3
given by
⌈ctx⌉ :=
c if x = 0
0 if x > 0
Note that the above homomorphism can be thought of as setting t1 = 0, whichis intuitive when t1 is thought of as infinitesimally small.
We shall use the following order on (0,∞)tR, which again is intuitive if t1 isthought of as being infinitesimally small and positive:
x1ty1 < x2t
y2 whenever y1 > y2 or y1 = y2 and x1 < x2
2.2 Exploded T fibrations
The following definition of abstract exploded fibrations is far too general, butit allows us to talk about local models for exploded torus fibrations as abstractexploded fibrations without giving too many definitions beforehand.
Definition 2.1. An abstract exploded fibration B consists the following:
1. A set of pointsp→ B
2. A Hausdorff topological space [B] with a map
p→ B [·]−→ [B]
3. A sheaf of Abelian groups on [B], E×(B) called the sheaf of exploded func-tions on B so that:
(a) Each element f ∈ E×(U) is a function defined on the set of pointsover U ,
f : [·]−1(U) −→ R∗tR
where R∗ denotes the multiplicatively invertible elements of some ringR. (For this paper, we shall use R = C.)
(b) E×(U) includes the constant functions if U 6= ∅.(c) Multiplication is given by pointwise multiplication in R∗tR.
(d) Restriction maps are given by restriction of functions.
We shall use the terminology of an open subset U of an abstract explodedfibration B to mean an open subset U ⊂ [B] along with the set of points in[·]−1(U) and the sheaf E×(U). The strange notation for the set of points in B,p→ B is used because it is equal to the set of morphisms of a point into B.
Definition 2.2. A morphism f : B −→ C of abstract exploded fibrations is amap on points and a continuous map of topological spaces so that the followingdiagram commutes
p→ B f−→ p→ C↓ [·] ↓ [·][B]
[f ]−−→ [C]
4
and so that f preserves E× in the sense that if g ∈ E×(U), then f g is inE×(
[f ]−1(U))
f∗g := f g ∈ E×(
[f ]−1(U))
Note that the reader should not attempt to work directly with the abovedefinition as it is too general. It simply enables us to talk about exampleswithout giving a complete definition beforehand. The next sequence of exampleswill give us local models for exploded T fibrations.
Example 2.3.
Any smooth manifold M is a smooth exploded T fibration. In this case, theunderlying toplogical space [M ] is just M with the usual topology, the set ofpoints p → M the usual set of points in M , and [·] : p → M −→ [M ] theidentity map. The sheaf E×(M) consists of all functions of the form f ta wheref ∈ C∞(M,C∗) and a is a locally constant R valued function.
The reader should convince themselves that this is just a strange way ofencoding the usual data of a smooth manifold, and that a exploded morphismbetween smooth manifolds is simply a smooth map. This makes the category ofsmooth manifolds a full subcategory of the category of exploded T fibrations.
The above example should be considered as a ‘completely smooth’ object.At the other extreme, we have the following ‘completely tropical’ object.
Example 2.4.
The exploded fibration Tn is best described using coordinates (z1, . . . , zn)where each zi ∈ E×(Tn). The set of points in Tn, p → Tn are then iden-tified with
(
C∗tR)n
by prescribing the values of (z1, . . . , zn). The underlying
topological space [Tn] is tRn
(given the usual topology on Rn), and
[(z1, . . . , zn)] = (⌊z1⌋, . . . , ⌊zn⌋) or [(c1ta1 , . . . , cntan)] = (ta1 , . . . , tan)
Note that there is a (C∗)n worth of points p −→ Tn over every topological point[p] ∈ [Tn].
Exploded functions in E× (Tn) can be written in these coordinates as
cty zα := cty∏
zαi
i
where c ∈ C∗, y ∈ R and α ∈ Zn are all locally constant functions.
The reader should now be able to verify the following observations:
1. A morphism from Tn to a smooth manifold is simply given by a map[Tn] −→M which has as its image a single point in M .
2. A morphism from a connected smooth manifold M to Tn has the infor-mation of a map [f ] from M to a point [p] ∈ [Tn] and a smooth map ffrom M to the (C∗)n worth of points over [p]. The map f is determinedby specifying the n exploded functions
f∗(zi) ∈ E× (M)
5
Note in particular that E× (M) is equal to the sheaf of smooth morphismsof M to T. This will be true in general. A smooth morphism f : B −→ Tn
is equivalent to the choice of n exploded functions in E× (B) corresponding tof∗(zi).
Now, a hybrid object, part ‘smooth’, part ‘tropical’.
Example 2.5.
The exploded fibration T1[0,∞) := T1
1 is more complicated. We shall describe
this using the coordinate z ∈ E×(T11). The set points p → T1
1 is identified
with C∗tR+
by specifying the value of z(p) ∈ C∗tR+
.The underlying topological space is given as follows:
[T11] :=
C∐
tR+
0 ∈ C = t0 ∈ tR+
:= (z, ta) ∈ C× tR+
so that za = 0
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
Our map from points to [T11] is given by
[z] := (⌈z⌉, ⌊z⌋)
The above expression uses the maps defined in section 2.1, ⌊cta⌋ = ta, and ⌈cta⌉is equal to c if a = 0 and 0 if a > 0. Note that there is one point p → T1
1
over points of the form (c, t0) ∈ [T11], a C∗ worth of points over (0, ta) ∈ [T1
1] fora > 0 and no points over (0, t0) ∈ [T1
1].This object should be thought of as a copy of C which we puncture at 0,
and consider this puncture to be an asymptoticaly cylindrical end. Each copyof C∗ over (0, ta) should be thought of as some ‘cylinder at ∞’. Note that eventhough there is a (0,∞) worth of two dimensional cylinders involved in thisobject, it should still be thought of being two dimensional. This strange featureis essential for the exploded category to have a good holomorphic curve theory.(Actually, we could just have easily had a Q+ worth of cylinders at infinity, and
6
worked over the semiring CtQ instead of CtR, but this author prefers the realnumbers.)
We shall use the notation
⌈T11⌉ :=
[T11]
tR+ = 0
= C
⌊T11⌋ :=
[T11]
C = 0= tR
+
We shall call ⌈T11⌉ the smooth or topological part of T1
1 and ⌊T11⌋ the tropical
part of T11.
We can write any exploded function h ∈ E×(
T11
)
as
h(z) = f(⌈z⌉)ty zα for f ∈ C∞(C,C∗), and y ∈ R, α ∈ Z locally constant.
We can now see that T11 is a kind of hybrid of the last two examples. Re-
stricting to an open set contained inside ⌊z⌋ = t0 ⊂ [T11] we get part of a
smooth manifold. Restricting to an open set contained inside ⌈z⌉ = 0 ⊂ [T11],
we get part of T.
We can define +E× (B) to consist of all functions in E×(B) which take values
only in C∗tR+
. Note that a smooth morphism f : B −→ T11 from any exploded
T fibration B that we have described in examples so far is given by a choice ofexploded function f∗(z) ∈ +E× (B). This will be true for exploded T fibrationsin general.
We can build up ‘exploded curves’ using coordinate charts modeled on opensubsets of T1
1. For an example of this see the picture on page 14. For an exampleof a map of this to Tn see example 2.39 on page 23. Those interested in tropicalgeometry may like the following example 2.40. For a local example of a familyof such objects, see example 2.47 on page 26.
We shall use the special notation Tmn to denote (T1
1)n × T(m−n). (These
spaces will be described rigorously below.) Before giving the most general localmodel for exploded T fibrations, we give the following definition which can bethought of as a local model for the tropical part of our exploded fibrations.
Definition 2.6. An integral cone A ⊂ tRm
is a subset of tRm
given by a collec-tion of inequalities
A :=
ta := (ta1 , . . . , tam) ∈ tRm
so that ta·αi
:=∏
j
(taj )αij ∈ tR
+
In other words, A := ta so that a · αi ≥ 0where αi are vectors in Zm.
The dual cone A∗ to A is defined to be
A∗ := α ∈ Zm : α · a ≥ 0, ∀ta ∈ A
7
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
The following is the most general local model for exploded T fibrations.
Example 2.7.
Given any integral cone A ⊂ tRm
which generates tRm
as a group, we willnow describe the exploded fibration Rn × Tm
A . We shall use coordinates
(x1, . . . , xn, z1, . . . , zm)
The set of points p→ Rn × TmA is then given by
(x1, . . . , xm, c1ta1 , . . . , cmtam) ∈ Rn ×
(
C∗tR)m
; (ta1 , . . . , tam) ∈ A
We will describe the underlying topological space [Rn × TmA ] using some
extra coordinate choices. Let α1, . . . , αk generate the dual cone A∗ ⊂ Zm asa semigroup with the relations
∑
i
ri,jαi = 0
Label coordinates for Ck as (zα1
, . . . , zαk
), and use ta to denote points in A.We can describe the underlying topological space, [Rn × Tm
A ] as a subspace
[Rn × TmA ] =
(x, zαi
, ta);∏
i
(zαi
)ri,j = 1, (αi · a)zαi
= 0
⊂ Rn × Ck ×A
Again we shall use the notation
⌊Rn × TmA ⌋ := A, the tropical part
⌈Rn × TmA ⌉ :=
∏
i
(zαi
)ri,j = 1
⊂ Rn × Ck, the smooth part
(This smooth part is equal to Rn times the toric variety corresponding to thecone A.) Using the shorthand zα :=
∏
i zαi
i , and recalling that ⌈cta⌉ is equal toc if a = 0, and equal to 0 if a > 0, we can write the map from the set of pointsdown to the smooth part as
⌈(x, z)⌉ := (x, ⌈zα1⌉, . . . , ⌈zαk⌉)
The map from the set of points down to the tropical part is given by
⌊(x, z)⌋ := (⌊z1⌋, . . . , ⌊zm⌋) or ⌊(x, c1ta1 , . . . , cmtam)⌋ = (ta1 , . . . , tam)
8
The map from the set of points down to the underlying topological space is thendefined by
[(x, z)] := (⌈(x, z)⌉, ⌊(x, z)⌋)(Note that we can also describe [Rn × Tm
A ] as the closure of the image of theabove map inside Rn × Ck ×A.) Exploded functions h ∈ E× (Rn × Tm
A ) are allfunctions that can be written as
h(x, z) := f (⌈(x, z)⌉) zαty
where f ∈ C∞(Rn × Ck,C∗), and α ∈ Zm and y ∈ R are locally constant.It is important to note that two exploded functions are equal if they have thesame values on points (which means that we can represent the same explodedfunction using different smooth functions f .) Note also that this definition isindependent of the choice of basis αi for A∗.
The earlier examples T11 = T1
t[0,∞) , and Tn = TntR
n , and
Tmn = (T1
1)n × T(m−n) := Tm
t(R+)n×R
(m−n)
These examples are much easier to understand than the most general model, soif the above definition was confusing, it is worthwhile reading through it againwith these examples in mind. In this case, we can choose the obvious basis forA∗ given by the first n standard basis vectors, so zα
i
:= zi. The smooth partof Tm
n , is then ⌈Tmn ⌉ := Cn with coordinates (⌈z1⌉, . . . , ⌈zn⌉). An important
example of exploded fibrations locally modeled on open subsets of Tmn is given
in example 2.37 on page 22. We need the more confusing models TmA because
they arise naturally in the intersection theory of exploded fibrations. They alsooccur in moduli spaces of holomorphic curves. We need to consider families ofholomorphic curves with total spaces locally modeled on these more complicatedTmA .
Definition 2.8. A smooth exploded T fibration B is an abstract exploded fibra-tion locally modeled on open subsets of Rn × Tm
A .In other words, each point in [B] has a neighborhood U which is isomorphic
as an abstract exploded fibration to some open set in Rn × TmA . The model
Rn × TmA used may depend on the open set.
There are some extra sheaves of functions which are defined on any smoothexploded T fibration B which will come in useful:
Definition 2.9. The sheaf of smooth functions C∞(B) is the sheaf of smoothmorphisms of B to R considered as a smooth exploded T fibration.
Definition 2.10. The sheaf of integral affine functions A×(B) consists of func-tions of the form ⌊f⌋ where f ∈ E×(B).
Recall that ⌊cta⌋ := ta. We shall also use the terminology that the orderof cta is a. The sheaf A×(Tm
A ) corresponds to functions on A of the formta 7→ ty+α·a where α ∈ Zm and y ∈ R.
9
Definition 2.11. The sheaf of exploded tropical functions E(B) is the sheaf ofCtR valued functions which are locally equal to a finite sum of exploded functionsE×. (‘Sum’ means sum using pointwise addition in CtR.) The sheaf of tropicalfunctions A(B) consists of functions of the form ⌊f⌋ for some f ∈ E(B).
For example A(Tn) consists of functions of the form tf where f : Rn −→ R
is some continuous, concave piecewise integral affine function. Functions ofthis form are usually called tropical functions. We shall usually not use E(B)because the addition is strange, and E×(B) is naturally defined as the sheaf ofmorphisms to T. The only place that E(B) will be used in this paper will be inthe definition of the tangent space of B. The operation of addition is neededhere to state the usual property of being a derivation. The other reason thataddition was mentioned in this paper was to emphasize the links with tropicalgeometry.
The inclusion ι : C −→ CtR defined by ι(c) = ct0 induces an inclusion offunctions
ι : C∞(B) −→ E(B)
ι(f)(p) := ι(f(p))
Definition 2.12. Using the notation of example 2.7, a Ck function on Rn×TmA ,
f ∈ Ck(Rn × TmA ) is a function on [Rn × Tm
A ] which is equal to a Ck function
applied to the variables xi and zαi
. A Ck exploded function f ∈ Ek×(Rn ×TmA )
is a function of the form f zαty where f is a Ck function taking values in C∗.We can define Ck morphisms of exploded fibrations to be morphisms of abstractexploded fibrations using the sheaf Ek× instead of E×
We can consider an arbitrary Hausdorff topological space to be a C0 explodedfibration. A continuous morphism of B to a Hausdorff topological space X isgiven by a map f : B −→ [B] −→ X so that the pullback of continuous realvalued functions on X are C0 functions on B. Note that in general, the map[·] : B −→ [B] is not a continuous morphism in this sense.
Definition 2.13. The topological or smooth part of B, ⌈B⌉ is the unique Haus-dorff topological space ⌈B⌉ with the following universal property: There is a C0
morphism
B⌈·⌉−−→ ⌈B⌉
so that any other C0 morphism from B to a Hausdorff topological space factorsuniquely through ⌈·⌉:
Given f : B −→ X
there exists a unique h : ⌈B⌉ −→ X
so that f = h ⌈·⌉
We can construct ⌈B⌉ as the image of B in RC0(B). The universal propertyof ⌈·⌉ makes it clear that taking the topological part of B is a functor from theexploded category to the category of Hausdorff topological spaces.
Definition 2.14. We shall use the adverb ‘topologically’ to indicate a propertyof ⌈B⌉. In particular,
10
1. A sequence of points pi −→ B converges topologically to p −→ B if theirimages converge in ⌈B⌉. In this case we use the notation ⌈pi⌉ → ⌈p⌉ orsay these points converge in ⌈B⌉. Note that this is equivalent to f(pi)converging to f(p) for every continuous R valued function f ∈ C(B).
2. The exploded fibration B is topologically compact if ⌈B⌉ is compact.
3. A map f : B −→ C is topologically proper if the induced map ⌈f⌉ : ⌈B⌉ −→⌈C⌉ is proper.
Note that limits in the above sense are of course not unique. A manifold istopologically compact when considered as an exploded fibration if and only if itis compact. Another example is any connected open subset of Tn. Sometimesthe topological part of B gives slightly misleading information about B, asthe following example is intended to demonstrate. We shall give a more usefulanalogue of ‘compactness’ in definition 2.19 shortly.
Example 2.15.
Our local model Rn × TmA has a smooth part ⌈Rn × Tm
A ⌉ which is equal tothe product of Rn with the toric variety corresponding to the integral affinecone A. The following two examples have ‘misbehaved’ smooth parts: Considerthe exploded fibration formed by quotienting R × T by (x, z) = (x + 1, z) and(x, z) = (x+ π, t1z). The smooth part of this is a single point, even though it ismade up of pieces which have smooth parts equal to a circle. Another exampleis given by the subset of R × T which is the union of the set where ⌊z⌋ < t0
with the set where ⌊z⌋ > t1 and x > 0. (Note that these sets overlap becauset0 > t1.) The smooth part of this is equal to R.
For many of our proofs, we shall restrict to a class of exploded fibrationsin which the smooth part and the tropical part of the exploded fibration givebetter information than the above two examples. To explain this class of ‘basic’exploded fibrations, we shall first construct some elementary examples.
Example 2.16.
Suppose that P ⊂ tRm
is a convex integral affine polygon with nonemptyinterior defined by some set of inequalities:
P := ta ∈ tRm
so that tci+a·αi ≤ t0, tci′+a·αi′
< t0
where αi ∈ Zm and ci ∈ R. This can simply be viewed as a convex polyhedronin Rm which has faces with rational slopes. View P as a stratified space in thestandard way, (so the strata of P consist of the vertices of P , the edges of Pminus the vertices of P , the two dimensional faces of P minus all vertices andedges, and so on, up to the interior of P ). Note that as some of the defininginequalities may be strict, P may be missing some faces or lower dimensionalstrata.
Then there is an exploded fibration TmP associated with P ⊂ tR
m
, which isconstructed by gluing together coordinate charts modeled on Tm
Akwhere Ak are
the integral affine cones corresponding to the strata of P .
11
Explicitly, suppose that the point tak ∈ tR
m
is inside the kth strata of P , and
Ak is the integral affine cone so that locally near tak
, P is equal to Ak shifted
by tak
.
near tak
, P = ta ∈ tRm
so that ta−ak ∈ AkThere are coordinates z1, . . . , zm ∈ E× (Tm
P ) so that the set of points in TmP
are given by specifying values (z1(p), . . . , zm(p)) ∈(
C∗tR)m
so that
(⌊z1(p)⌋, . . . , ⌊zm(p)⌋) ∈ P
Denote by Pk ⊂ P the subset of P which is the union of all strata of P whoseclosure contains the kth strata of P . There is a corresponding subset Tm
Pkof
TmP :
TmPk
:= z so that ⌊z⌋ ∈ PkWe put an exploded structure on Tm
Pkso it is isomorphic to the corresponding
subset of TmAk
with the standard coordinates given by (t−ak1 z1, . . . , t
−akm zm). It
is an easy exercise to check that the restriction of the exploded structure thisgives to Tm
Pk∩ Tm
Pk′is compatible with the restriction of the one coming from
TmPk′
, so this gives a globally well defined exploded structure.An alternative way to describe this structure is as follows: Consider the
collection of functions w := tczα so that on P , ⌊w⌋ ≤ t0. Choose some fi-nite generating set w1, . . . , wn of these functions so that any other func-
tion of this type w can be written as w = tcwβ1
1 · · · wβnn where βi ∈ N and
c ∈ [0,∞). Exploded functions on TmP can then be described as functions of the
form f(⌈w1⌉, . . . , ⌈wn⌉)tczα, where f : Cn −→ C∗ is smooth. The underlyingtopological space [Tm
P ] is equal to the closure of the set
⌈w1⌉, . . . , ⌈wn⌉, ⌊z1⌋, . . . , ⌊zm⌋ ⊂ Cn × P
The tropical part of TmP is given by
⌊TmP ⌋ = P = ⌊z1⌋, . . . , ⌊zm⌋ ⊂ tR
m
The smooth part of TmP is given by
⌈TmP ⌉ := ⌈w1⌉, . . . , ⌈wn⌉ ⊂ Cn
Simple examples of the above construction are T1(a,b) which is equal to the
subset of T where ta > ⌊z⌋ > tb, and T1[0,b) which is equal to the subset of T1
1
where ⌊z⌋ > tb. (Strictly speaking, we should be writing T1t(a,b) , as the above
confuses the interval (a, b) ⊂ R with t(a,b) ⊂ tR.)
Similarly, we can construct Rk×TmP . This has coordinates (x, z) ∈ Rk×Tm
P ,has underlying topological space [Rk×Tm
P ] = Rk×[TmP ], so that [(x, z)] = (x, [z]),
and has exploded functions equal to functions of the form f(x, ⌈w1⌉, . . . , ⌈wn⌉)tczα,where f : Rk × Cn −→ C∗ is smooth.
Example 2.17.
12
Given a smooth manifold M and m complex line bundles on M , we canconstruct the exploded fibration M ⋊ Tm
P as follows: Denote by E the corre-sponding space of m C∗ bundles over M . This has a smooth free (C∗)
maction.
The exploded fibration TmP also has a (C∗)
maction given by multiplying the
coordinates z1, . . . , zm by the coordinates of (C∗)m. M ⋊ Tm
P is the explodedfibration constructed by taking the quotient of E×Tm
P by the action of (C∗)m by(c, c−1). As the action of (C∗)
mis trivial on ⌊Tm
P ⌋, the tropical part of M ⋊TmP
is still defined, and is equal to P .Alternately, choose coordinate charts on E equal to U × (C∗)m ⊂ Rn × Cn.
The transition maps are of the form (u, z1, . . . , zm) 7→ (φ(u), f1(u)z1, . . . , fm(u)zm).Replace these coordinate charts with U ×Tm
P ⊂ Rn×TmP , and replace the above
transition maps with maps of the form (u, z1, . . . , zm) 7→ (φ(u), f1(u)z1, . . . , fm(u)zm).The map to the tropical part ⌊M ⋊ Tm
P ⌋ := P in these coordinates is given by
⌊(u, z1, . . . , zn)⌋ = (⌊z1⌋, . . . , ⌊zn⌋) ∈ P
Definition 2.18. An exploded fibration B is basic if
1. There exists a Hausdorff topological space ⌊B⌋ called the tropical part ofB, along with a map
⌊·⌋ : p→ B −→ [B] −→ ⌊B⌋
2. The space ⌊B⌋ is a union of strata ⌊Bi⌋ so that
(a) Each strata ⌊Bi⌋ ⊂ ⌊B⌋ is an integral affine space equal to some
open convex integral affine polygon in tRk
of the form considered inexample 2.16. (The dimension k depends on the strata.)
(b) The closure of ⌊Bi⌋ ⊂ ⌊B⌋ is a union of strata which is also equalto an integral affine polygon ⌊Bi⌋ of the form considered in example2.16 with some strata identified.
3. Use the notation Bi for inverse image of ⌊Bi⌋ under the map ⌊·⌋, and callthis a strata of B.
(a) Each strata Bi has a smooth part ⌈Bi⌉ which is a smooth manifold.
(b) Each strata Bi is equal to an exploded space ⌈Bi⌉⋊ Tk⌊Bi⌋
using the
construction of example 2.17. The map ⌊·⌋ restricted to Bi is themap ⌊·⌋ : ⌈Bi⌉⋊ Tk
⌊Bi⌋−→ ⌊Bi⌋ from example 2.17.
(c) There is a neighborhood of each strata Bi which is open in ⌈B⌉ whichis isomorphic to some subset of ⌈Bi⌉⋊Tk
⌊Bi⌋which is a topologically
open neighborhood of ⌈Bi⌉⋊ Tk⌊Bi⌋
⊂ ⌈Bi⌉⋊ Tk
⌊Bi⌋.
It is an easy exercise to check that if B is basic, the smooth part ⌈B⌉ is alsoa stratified space, with strata equal to the manifolds ⌈Bi⌉. The stratificationof ⌈B⌉ is in some sense dual to the stratification of ⌊B⌋ in that ⌊Bi⌋ is in theclosure of ⌊Bj⌋ if and only if ⌈Bj⌉ is in the closure of ⌈Bi⌉.
All our examples up until now apart from those introduced in example 2.15have been basic. Another example of an exploded fibration that is not basic isgiven by quotienting R×T by (x, z) = (x+1, z−1). A less pathological example
13
of an exploded fibration that is not basic is given by the quotient of T by z = t1z.Any exploded fibration is locally basic.
To summarize the topological spaces we have involved in the case where B
is basic, we have the following topological spaces.
p→ B −→[B] −→ ⌈B⌉↓⌊B⌋
1. The smooth or topological part ⌈B⌉. Any continuos morphism of B to aHausdorff topological space factors through ⌈B⌉. The smooth part of asmooth manifold M is M . The smooth part of Tn is a single point.
2. The tropical part ⌊B⌋. This is a stratified integral affine space, which canbe thought of as what B looks like on the large scale. The tropical partof a connected smooth manifold is a single point. The tropical part of Tn
is tRn
.
3. The underlying topological space [B]. This strange topological space isa mixture of ⌈B⌉ and ⌊B⌋. It is the topological space on which E× is asheaf, and this is the topology in which B is locally modeled on Rn×Tm
A .
4. In the case that B is not a manifold, the set of points p → B is notHausdorff in any of the above topologies. We can put a topology on thisset by choosing the strongest topology in which all continuous morphismsfrom manifolds into B are continuous maps to p→ B. For examplep → T1
1 is homeomorphic to the disjoint union of [0,∞) copies of C∗.(When we have defined tangent spaces and metrics on exploded fibrations,this would be the topology coming from any metric.)
The following is a picture of an ‘exploded curve’ built up using coordinatecharts modeled on open subsets of T1
1. Note that ⌈C⌉ is what we would see ifwe mapped C to a smooth manifold. On the other hand, we would see ⌊C⌋ if wemapped C to Tn.
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]
⌈C⌉
⌊C⌋
14
Definition 2.19. An exploded T morphism f : B −→ C is complete if thefollowing conditions hold
1. f is topologically proper.
2. Any mapg : T1
(0,l) −→ B
extends to a continuous map T1[0,l) −→ B if and only if the map f g
extends to a continuous map T1[0,l) −→ C.
We say that an exploded fibration B is complete if the map from B to a pointis complete.
The notation T1(0,l) is that introduced in example 2.16. An equivalent way
of stating the second condition is that the map [g] : (0, l) −→ [B] extends to acontinuous map [0, l) −→ [B] if and only if [f g] : (0, l) −→ [C] extends to acontinuous map [0, 1) −→ [C]
Complete maps should be thought of as being analogous to proper maps.A smooth manifold is complete if and only if it is compact. An example of anon-compact exploded fibration which is complete is Tn. In the case that B isbasic, completeness means that the topological part ⌈B⌉ is compact, and thetropical part ⌊B⌋ is a complete stratified integral affine space.
An interesting example of an exploded fibration which is topologically com-pact but not complete is given by the following: Let M ×T be the product of acompact manifold with T. This is complete, but we can construct an interestingnon complete example as follows: choose an open subset U ⊂ M . Considerthe subset of M × T given by the union of U × T1
(−∞,1) with M × T1(0,∞). The
smooth part of this is the compact manifold M , but it is not complete.An example of an interesting exploded fibration which is complete but not
basic is given by the quotient of R× T by (x, z) = (x + 1, t1z).
The following lemma is easy to prove:
Lemma 2.20. 1. if f and g are complete, then f g is complete.
2. if f g is complete, then g is complete.
2.3 Tangent space
Definition 2.21. A smooth exploded vectorfield v on B is determined by maps
v : C∞(B) −→ C∞(B)
v : E(B) −→ E(B)
so that
1.v(f + g) = v(f) + v(g)
2.v(fg) = v(f)g + fv(g)
15
3.v(ctyf) = ctyv(f)
4. The action of v is compatible with the inclusion ι : C∞ −→ E in the sensethat
v(ιf) = ιv(f)
Smooth exploded vectorfields form a sheaf. The action of the restriction of vto U on the restriction of f to U is the restriction to U of the action of v on f .
We can restrict a vector field to a point p −→ B to obtain a tangent vectorat that point. This is determined by maps
v : C∞(B) −→ R
v : E(B) −→ CtR
satisfying the above conditions with condition 2 replaced by
v(fg) = v(f)g(p) + f(p)v(g)
Denote by TpB the vector space of tangent vectors over p −→ B.
We can add smooth exploded vectorfields and multiply them by smoothreal valued functions. We shall show below that the sheaf of smooth explodedvectorfields is equal to the sheaf of smooth sections of TB, which is a real vectorbundle over B.
Lemma 2.22. Differentiation does not change the order of a function in thesense that given any smooth exploded vectorfield v and exploded function f ∈ E,
⌊f⌋ = ⌊vf⌋Proof:
vt0 = v(ι1) = 0t0
Now we can apply v to the equation f = 1t0f , so vf = 0t0f + 1t0vf . Taking⌊·⌋ of this equation gives
⌊vf⌋ = ⌊f⌋+ ⌊vf⌋ i.e. ⌊vf⌋ ≥ ⌊f⌋(Recall that we use the order tx < ty if x > y as we are thinking of t as beingtiny. So tx + ty = tx means that tx ≥ ty.) Now,
vta = v(tat0) = ta0t0 = 0ta
Now suppose that at some point ⌊vf⌋ > ⌊f⌋. Then we can restrict to a smallneighborhood of this point, and choose an a so that ⌊vf⌋ > ta > ⌊f⌋. But thenf + ta = ta, but ⌊v(f + ta)⌋ = ⌊vf⌋+ ⌊ta⌋ = ⌊vf⌋, which is a contradiction, andthe lemma is proved.
Lemma 2.23. For any smooth exploded fibration B, there exists a smooth ex-ploded fibration TB, the tangent space of B, along with a canonical smoothprojection π : TB −→ B that makes TB into a real vector bundle over B. Thesheaf of smooth exploded vectorfields is equal to the sheaf of smooth sections ofthis vector bundle.
In particular,T (Rn × Tm
A ) = R2n+2m × TmA
16
Proof:We shall first prove that T (Rn × Tm
A ) = R2n+2m × TmA . We shall use co-
ordinate functions xi for Rn and zi for TmA , as in example 2.7 on page 8. A
section of R2n+2m × TmA −→ Rn × Tm
A is given by n+ 2m smooth functions onRn × Tm
A . To a vectorfield v, associate the n smooth functions v(xi), and them smooth complex valued functions given by ⌈v(zi)z−1
i ⌉. (Lemma 2.22 tellsus that ⌊z−1
i v(zi)⌋ = t0, so this makes sense.) This gives us n + 2m smoothreal valued functions, and therefore gives us a section to associate with ourvectorfield.
Now we must show that the arbitrary choice of n+2m smooth functions onTmA to equal v(xi) and ⌈v(zi)z−1
i ⌉ will uniquely determine a smooth explodedvectorfield.
First, recall that exploded functions are a sum of functions of the form f tyzα,where f is some smooth function of x and zα
j
. Use the notation
zαj
:= etαj+iθαj
There is the following formula for v(f) which follows from the axioms:
v(f) =∑
v(xi)∂f
∂xi
+∑
i,j
ℜ(
v(zi)z−1i
)
αji
∂f
∂tαj
+ ℑ(
v(zi)z−1i
)
αji
∂f
∂θαj
Note that this is well defined, despite the fact that ∂f∂t
αjand ∂f
∂θαj
are not
necessarily well defined. Note also that this is a smooth function, and is realvalued if f is real valued. The corresponding formula for an exploded functionis
v
(
∑
α
fαtyα zα
)
:=∑
α
(
v(fα) + fα∑
i
αiv(zi)z−1i
)
tyα zα
It can be shown that v satisfying such a formula satisfies all the axioms forbeing a smooth exploded vectorfield, is well defined, and is zero if and only ifv(xi) = 0 and ⌈v(zi)z−1
i ⌉ = 0.This shows that TRn × Tm
A = R2n+2m × TmA . The lemma then follows from
our coordinate free definition of a smooth vectorfield, and the fact that everysmooth exploded T fibration is locally modeled on exploded fibrations of thistype.
For any smooth exploded T fibrationB, we now have that TB is a real vectorbundle. In local coordinates xi, zj, a basis for this vectorbundle is given by ∂∂xi and the real and imaginary parts of zi ∂
∂zi. The dual of this bundle T ∗B
is the cotangent space. A basis for this is locally given by dxi and the realand imaginary parts of z−1
i dzi. We can take tensor powers (over smooth realvalued functions) of these vector bundles, to define the usual objects found onsmooth manifolds. For example, a metric on B is a smooth, symmetric, positivedefinite section of T ∗B⊗ T ∗B.
Definition 2.24. An integral vector v at a point p −→ B is a vector v ∈ TpB
so that for any exploded function f ∈ E×(B),
v(f)f−1 ∈ Z
Use the notation ZTpB ⊂ TpB to denote the integral vectors at p −→ B.
17
For example, a basis for ZTTn is given by the real parts of zi∂∂zi
. The onlyintegral vector on a smooth manifold is the zero vector.
Given a smooth morphism f : B −→ C, there is a natural smooth morphismdf : TB −→ TC which is the differential of f , defined as usual by
df(v)g := v(g f)
Of course, df takes integral vectors to integral vectors.
2.4 Ck,δ regularity
In this section, we define the topology in which we shall prove our compactnesstheorem for holomorphic curves. The regularity C∞,δ is the level of regularitythat the moduli stack of holomorphic curves can be expected to exhibit. A C∞,δ
function is a generalization of a function on a manifold with a cylindrical endwhich is smooth on the interior, and which decays exponentially along with allits derivatives on the cylindrical end. The reader wishing a simple introductionto holomorphic curves in exploded T fibrations may skip this somewhat technicalsection on first reading.
The following is a definition of a strata of an integral affine cone, similar tothe definition of strata given in definition 2.18 on page 13.
Definition 2.25. The closure of a strata of an integral cone A := ta ; a ·αi ≥0 is a subset of A defined by the vanishing of a · α for some α ∈ A∗. A strataS of A is a subset which is equal to the closure of a strata minus the closureof all strata of smaller dimension. The zero substrata is the substrata on whicha · α = 0 for all α ∈ A∗.
Definition 2.26. Given any C0 function f defined on a strata S ⊂ A in Rn ×TmA , define
eS(f)(x, z) := f(x, zta)
where ta := (ta1 , . . . , tam) is any point in S, and zta means (z1ta1 , . . . , zmtam).
For example, T22 := T2
[0,∞)2 has two one dimensional strata
S1 := ⌊z2⌋ = t0, ⌊z1⌋ 6= t0 S2 :=
⌊z1⌋ = t0, ⌊z2⌋ 6= t0
If we have a function f ∈ C0(T22), then
eS1f(z1, z2) = f(0, z2) eS2f(z1, z2) = f(z1, 0)
Note that the operations eSicommute and eSi
eSi= eSi
.
Definition 2.27. If I denotes any collection of strata S1, . . . , Sn, we shalluse the notation
eIf := eS1 (eS2 (· · · eSnf))
∆If :=
(
∏
Si∈I
(id−eSi)
)
f
18
For example,
∆S1,S2f(z1, z2) := (1− eS1)(1 − eS2)(f)(z1, z2)
:= f(z1, z2)− f(0, z2)− f(z1, 0) + f(0, 0)
Note that if S ∈ I, eS∆I = 0. In the above example, this corresponds to∆S1,S2f(z1, 0) = 0 and ∆S1,S2f(0, z2) = 0.
We shall need a weight function wI for every collection of nonzero strataI. This will have the property that if f is any smooth function w−1
I ∆If willbe bounded on any topologically compact subset of Rn × Tm
A (in other words,a subset with a compact image in the topological part ⌈Rn × Tm
A ⌉). Considerthe ideal of functions of the form zα so that ∆I⌈zα⌉ = ⌈zα⌉ (In other words,
eS⌈zα⌉ = 0 for all S ∈ I.) Choose some finite set of generators zαi for thisideal. Then define
wI :=∑
∣
∣
∣⌈zαi⌉
∣
∣
∣
Continuing the example started above, we can choose wS1 = |z1|, wS2 = |z2|,wS1,S2 = |z1z2|, and wS1∩S2 = |z1|+ |z2|.
Note that given any other choice of generators, the resulting w′I is bounded
by a constant times wI on any topologically compact subset of Rn × TmA . Note
also that wIwI′ is bounded by a constant times wI∪I′ on any topologicallycompact subset of Rn × Tm
A .
We shall now define Ck,δ for any 0 < δ < 1:
Definition 2.28. Define C0,δ to be the same as C0. A sequence of smoothfunctions fi ∈ C∞(Rn×Tm
A ) converge to a continuous function f in Ck,δ(Rn×TmA ) if the following conditions hold:
1. Given any collection of at most k nonzero strata I,
∣
∣w−δI ∆I(fi − f)
∣
∣
converges to 0 uniformly on compact subsets of ⌈Rn × TmA ⌉ as i → ∞.
(This includes the case where our collection of strata is empty and fi → funiformly on compact subsets.)
2. For any smooth exploded vectorfield v, v(fi) converges to some functionvf in Ck−1,δ.
Define Ck,δ(Rn × TmA ) to be the closure of C∞ in C0 with this topology.
Define C∞,δ to be the intersection of Ck,δ for all k.
In particular, Ck ⊂ Ck,δ for 0 < δ < 1. Functions in Ck,δ can be thought ofas functions which converge a little slower than Ck functions when they approachdifferent strata. Thinking of a single strata as being analogous to a cylindricalend, this is similar to requiring exponential convergence (with exponent δ) onthe cylindrical end. We shall now start showing that we can replace smoothfunctions with C∞,δ functions in the definition of exploded torus fibrations tocreate a category of C∞,δ exploded torus fibrations.
19
Definition 2.29. A Ck,δ exploded function f ∈ Ek,δ,×(Rn × TmA ) is a function
of the form
f(x, z) := g(x, z)zαta where g ∈ Ck,δ (Rn × TmA ,C∗) , α ∈ Zm, ta ∈ tR
A sequence of exploded functions gizαtai converge in C∞,δ if the sequence offunctions gi does, and the sequence ai is eventually constant.
A Ck,δ exploded fibration is an abstract exploded fibration locally modeled onopen subsets of Rn × Tm
A with the sheaf Ek,δ,×.
The following observations are easy to prove:
Lemma 2.30. Ck,δ is an algebra over C∞ for any 0 < δ < 1.
Lemma 2.31. Given any ‘linear’ map
α : Rn × TmA −→ Rn′ × Tm′
B
α(x, z) :=(
Mx, zα1
, . . . , zαm′)
where M is a n by n′ matrix and αij is a m by m′ matrix with Z entries, α
preserves C∞,δ in the sense that given any function f ∈ C∞,δ(Rn′ × Tm′
B ),
α f ∈ C∞,δ (Rn × TmA )
More difficult is the following:
Lemma 2.32. Given any Ck,δ section of T (Rn × TmA ), we can define a map of
the form
exp(f)(x, z1, . . . , zm) :=(
x+ fRn(x, z), ef1(x,z)z1, . . . , efm(x,z)zm
)
If h is in Ck,δ, then h exp f is.
Proof:We must show that expressions of the following form decay appropriately.
∆I(h exp f)
Note thateS(h exp f) = (eSh) exp(eSf)
A little thought shows that we can rewrite our above expression in the followingform
∆I(h exp f) =∑
I′∪I′′=I,I′∩I′′=∅
((eI′∆I′′h) exp ∆I′) f
As an example for interpreting the notation above, we write
(h exp ∆S) f := h exp f − h exp(eSf)
as opposed toh exp (∆Sf) := h exp(f − eSf)
20
We have that because f is bounded on compact sets,
(∆Ih) f decays appropriately.
We can bound expressions of the form
((∆I′h) exp ∆I′′) f
by the size of the first |I ′′| derivatives of ∆I′h times the sum of all products ofthe form
∏ |∆Iif | where I ′′ is the disjoint union of Ii. We therefore get thatall of the terms in the above expression decay appropriately. The appropriatedecay of the derivatives of h exp f follows from the fact that C∞,δ is closedunder multiplication, and a similar argument to the above one.
Consider a map for which the pull back of exploded coordinate functions isin Ek,δ,× and the pullback of real coordinate functions is Ck,δ. Any map of thisform factors as a composition of maps in the form of Lemma 2.31 and Lemma2.32; (first a map in the form of Lemma 2.31 to the product of the domain andtarget, then a map of the form of Lemma 2.32, then a projection to the target,which is of the form of Lemma 2.31.) We therefore have the following:
Corollary 2.33. A morphism is Ck,δ if and only if the pull back of explodedcoordinate functions are Ek,δ,× functions, and the pull back of real coordinatefunctions are Ck,δ functions.
We shall need the following definition of convergence.
Definition 2.34. A sequence of Ck,δ exploded maps f i : A −→ B converges tof : A −→ B in Ck,δ if the pullback under f i of any local coordinate function onB converges in Ck,δ to the pullback under f .
2.5 Almost complex structures
Definition 2.35. An almost complex structure J on a smooth exploded T fi-bration B is an endomorphism of TB given by a smooth section of TB⊗ T ∗B
which squares to become multiplication by −1, so that for any coordinate chartmodeled on an open subset of Rn×Tm J of the real part of zi
∂∂zi
is the imaginarypart.
An almost complex structure J is a complex structure if there exist localcoordinates z ∈ Tm
A so that for all vectorfields v, iv(zj) = (Jv)(zj).
This differs from the usual definition of an almost complex structure onlyin the extra requirement that in standard coordinates, (Jv)zj = ivzj which isonly required to hold when zj is locally a coordinate on T. For instance, onT11, this requirement is only valid when ⌊z⌋ < t0. This extra requirement makes
holomorphic curves C∞,δ exploded T morphisms, and makes the tropical part ofholomorphic curves piecewise linear one complexes. If it did not hold, then wewould need to use a different version of the exploded category using R∗ insteadof C∗ to explore holomorphic curve theory. The analysis involved would besignificantly more difficult.
The following assumption allows us to use standard (pseudo)holomorphiccurve results.
21
Definition 2.36. An almost complex structure J on B is civilized if it inducesa smooth almost complex structure on the smooth part of B. This means thatgiven any local coordinate chart modeled on Tm
A , if we consider TmA as a subset
of Tm′
n defined by setting some monomials equal to 1, there exists an almostcomplex structure J on Tm′
n which induces a smooth almost complex structureon ⌈Tm′
n ⌉ = Cn so that the subset corresponding to TmA is holomorphic, and the
restriction of J is J .
The word civilized should suggest that our almost complex structure is wellbehaved in a slightly unnatural way. Any complex structure is automaticallycivilized, and there are no obstructions to modifying an almost complex struc-ture to civilize it. If we assume our almost complex structure is civilized, thenholomorphic curves are smooth maps. Otherwise, they will just be C∞,δ for anyδ < 1.
Example 2.37.
Suppose that we have a complex manifold M along with a collection of im-mersed complex submanifolds Ni so that Ni intersect themselves and each othertransversely. Then there is a smooth complex exploded fibration Expl(M) calledthe explosion of M . (There is actually a functor Expl from complex manifoldswith a certain type of log structure which is defined by the submanifolds Ni tothe category of exploded fibrations.)
We define Expl(M) as follows: choose holomorphic coordinate charts on Mwhich are equal to balls inside Cn, so that the image of the submanifolds Ni areequal to the submanifolds zi = 0. Then replace a coordinate chart U ⊂ Cn
by a coordinate chart U in Tnn with coordinates zi so that
U := z so that ⌈z⌉ ∈ U
Define transition functions as follows: the old transition functions are all ofthe form fi(z) = zjg(z) where g is holomorphic and non vanishing. Replace
this with fi(z) = zjg(⌈z⌉), which is then a smooth exploded function. Thisdefines transition functions which define Expl(M) as a smooth complex explodedfibration.
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M ⌊ExplM⌋
Definition 2.38. An exploded T curve is a 2 real dimensional, complete ex-ploded T fibration with a complex structure j.
A holomorphic curve is a holomorphic map of an exploded T curve to analmost complex exploded fibration.
22
A smooth exploded curve is a smooth map of an exploded T curve to a ex-ploded T fibration.
By a smooth component of a holomorphic curve C, we shall mean a connectedcomponent of the set of points in C which have a neighborhood modeled on anopen subset of C. By an edge of C we shall mean connected component in [C]minus the image of all smooth components. We shall call edges that have onlyone end attached to a smooth component ‘punctures’. Each smooth componentis a punctured Riemann surface with punctures corresponding to where it isconnected to edges. The information in a holomorphic curve C is equal tothe information of a nodal Riemann surface plus gluing information for eachnode parametrized by C∗t(0,∞) (for more details of this gluing information seeexample 2.47 on page 26).
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]
⌊f(C)⌋ ⊂ ⌊Tn⌋
∑
df(vi) = 0
Example 2.39.
Consider a smooth curve f : C −→ Tn. This is given by n exploded functionsf∗(z1), . . . , f
∗(zn) ∈ E×(C). Each smooth component of C is sent to the (C∗)n
worth of points over a particular point in the tropical part ⌊Tn⌋. We can chooselocal holomorphic coordinates w ∈ T1
1 on C. In these coordinates,
f(w) = (g1(⌈w⌉)ta1wα1 , . . . , gn(⌈w⌉)tanwαn) gi ∈ C∞(C,C∗), α ∈ Zn
(Our smooth curve is holomorphic if and only if the functions gi in our localcoordinate representation above are holomorphic.) In particular, f restrictedto a smooth component gives a smooth map of the corresponding puncturedRiemann surface to (C∗)n, and the homology class in H1 ((C
∗)n) of a looparound the puncture corresponding in the above coordinates to ⌈w⌉ = 0 is givenby α. Of course, the sum of all such homology classes from punctures of asmooth component is zero. As α also determines the derivative of f on edges,this can be viewed as some kind of conservation of momentum condition for thetropical part of our curve, ⌊f⌋. In tropical geometry, this is called the balancingcondition.
Example 2.40.
One way to consider the image of some holomorphic curves in Tn is as the‘locus of noninvertablity’ of some set of polynomials
Pi(z) :=∑
ci,αzα i = 1, . . . , n− 1
23
We can consider the set
ZPi :=
z so that Pi(z) ∈ 0tR ∀i
Suppose that for all points p −→ ZPi, the differentials dPi at p are linearlyindependent. Then ZPi is the image of some holomorphic curve.
Let us examine the set ZPi more closely. For any point z0, denote by Si,z0
the set of exponents α so that ⌊Pi(z0)⌋ = ⌊ci,αzα⌋. Then there exists someneighborhood of ⌊z0⌋ in ⌊Tn⌋ so that
Pi =∑
α∈Si,z0
ci,αzα
The points inside ZPi over ⌊z0⌋ are then given by solutions of the equations
∑
α∈Si,z0
ci,αzα = 0 where ci,α = ci,α⌊ci,α⌋ and z = z⌊z⌋
Note that the above equation has solutions for z ∈ (C∗)n if and only if Si,z0 hasmore than 1 element. This corresponds to the tropical function ⌊Pi⌋ (which iscontinuous, piecewise integral affine, and convex) not being smooth at ⌊z0⌋. Wetherefore have that ⌊ZPi⌋ is contained in the intersection of the non-smoothlocus of the tropical polynomials ⌊Pi⌋.
2.6 Fiber product, refinements
Definition 2.41. Two smooth (or Ck,δ) exploded morphisms
Af−→ C
g←− B
are transverse if for every pair of points p1 −→ A and p2 −→ B so that f(p1) =g(p2), df(Tp1A) and dg(Tp2B) span Tf(p1)C.
Definition 2.42. If f and g are transverse smooth (or Ck,δ) exploded mor-phisms,
Af−→ C
g←− B
The fiber product A f×g B is the unique smooth (or Ck,δ) exploded T fibrationwith maps to A and B so that the following diagram commutes
A f×g B −→ A
↓ ↓B −→ C
and with the usual universal property that given any commutative diagram
D −→ A
↓ ↓B −→ C
there exists a unique morphism D −→ A f×g B so that the following diagramcommutes
D → A
↓ ց ↑B ← A f×g B
24
The existence of the fiber products defined above is left as an exercise. Theexistence in some special cases was proved in [12].
Example 2.43.
Consider the map f : TmA −→ Tn given by
f(z) = (zα1
, . . . , zαn
)
Denote by α the m× n matrix with entrys αij . The derivative of f is surjective
if α : Rm −→ Rn is. Denote by |α| ∈ N the size of α1 ∧ · · · ∧ αn ∈ ∧n(Zm).
(In other words, |α| ∈ N is the smallest number so that the above wedge is atmost |α| times any nonzero element of
∧n(Zm).) The fiber product of f with
the point (1, . . . , 1) corresponds to the points in TmA so that zα
i
= 1 for all i.
This is then equal to |α| copies of Tm−ntker α∩A
where we identify tRm−n
= tkerα.
The following should not be difficult to prove.
Conjecture 2.44. If f and g are complete and transverse
Af−→ C
g←− B
then the fiber product A f×g B −→ C is complete.
Definition 2.45. A family of exploded T fibrations over F is a map f : C −→ F
so that:
1. f is complete
2. for every point p −→ C,
df : TpC −→ Tf(p)F is surjective
and df : ZTpC −→ ZTpF is surjective
(The definition of complete is found on page 15. Recall also that integervectors in ZTpC are the vectors v so that for any exploded function, vf is aninteger times f . For example on T the integer vectors are given by integermultiples of the real part of z ∂
∂z, so the map T −→ T given by z2 is not a family
as it does not obey the last condition above.)The local normal form for coordinate charts on a smooth family is a map
TmA −→ Tk
B given by (z, w) 7→ z, where the cone B is given by the projection ofA to the first k coordinates, and the projection of every strata of A is a strataof B. This may differ from a product because the cone A may not be a productof B with something.
Example 2.46.
We can represent the usual compactified moduli space of stable curves Mg,n
as a complex orbifold. There exist local holomorphic coordinates so that theboundary of Mg,n in these coordinates looks like zi = 0. As in example 2.37on page 22, we can replace these coordinates zi with zi to obtain a complex
25
orbifold exploded fibration Expl(
Mg,n
)
. The forgetful map π : Mg,n+1 −→Mg,n induces an map
π : Expl(
Mg,n+1
)
−→ Expl(
Mg,n
)
This is a family, and each stable exploded curve with genus g and n markedpoints corresponds to the fiber over some point p −→ Expl
(
Mg,n
)
. This ex-ample is considered in a little more detail in [12]. (Obviously, it would take alittle more work to prove that ExplMg,n actually represents the moduli stackof stable exploded curves, but that is not the subject of this paper.)
Example 2.47.
The following example contains all interesting local behavior of the aboveexample. It is not a family only because it fails to be complete (it is nottopologically proper). Consider the map
π : T22 −→ T1
1 given by π∗z = w1w2
The derivative is surjective, as can be seen by the equation
π∗(z−1dz) = w−11 dw1 + w−1
2 dw2
The fibers of this map over smooth points z = ct0 are smooth manifolds equalto C∗ considered just as a smooth manifold with coordinates w1 and w2 ∈ C∗
related by w1w2 = c. (Note that there is no point with z = 0t0.)In contrast, the fibers of this map over points z = ctx with x > 0 are not
smooth manifolds. They can be described using coordinates w1, w2 ∈ T1[0,x) ⊂
T11, and transition maps give by w2w2 = ctx on their intersection, which is
T1(0,x). The picture below shows some of the fibers of this map
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0
26
Definition 2.48. A refinement of B is an exploded T fibration B′ with a mapf : B′ −→ B so that
1. f is complete
2. f gives a bijection between points in B′ and B
3. df is surjective
Example 2.49.
All refinements are locally of the following form: Suppose that the integralaffine cone A ⊂ tR
m
is equal to the union of a collection of cones Ai so thatthe intersection of any two of these Ai has codimension at least 1. Then usingthe standard coordinates (x, z) from example 2.7 (on page 8) on Rn × Tm
Aifor
each Ai, we can piece the coordinate charts Rn×Tm
Aitogether using the identity
map as transition coordinates. In these coordinates, the map down to Rn×TmA
is again the the identity map in these coordinates. (The fact that all refinementsare locally of this form can be proved using the observation that the pull backof coordinate functions on B must locally be coordinate functions on B′.) Soin the case that B is basic, a refinement is simply determined by a subdivisionof the tropical part ⌊B⌋. The effect on the smooth part ⌈B⌉ should remind thereader of a toric blowup.
The following lemma follows from the above standard local form for refine-ments.
Lemma 2.50. Given a refinement of B,
f : A −→ B
Any smooth exploded vector field v on B lifts uniquely to a smooth explodedvector field v on A so that df(v) = v.
The above lemma tells us that any smooth tensor field (such as an almostcomplex structure or metric) lifts uniquely to a smooth tensor field on anyrefinement. The above normal form implies that given any morphism C −→ B
and a refinement B′ −→ B, there exists a refinement C′ −→ C and a morphismC′ −→ B′ lifting the above map.
Definition 2.51. Call a holomorphic curve stable if it has a finite numberof automorphisms, and is not a nontrivial refinement of another holomorphiccurve.
If B has an almost complex structure, there is a bijection between stableholomorphic curves in B and stable holomorphic curves in any refinement B′.In fact, when the moduli space of stable holomorphic curves in B is smooth, themoduli space of stable holomorphic curves in B′ is a refinement of this modulispace.
27
2.7 Symplectic structures
Our definition of symplectic structures below should be regarded as provisional.Our definitions are enough to tame holomorphic curves, which is all that isrequired for this paper, but there is probably a better theory of symplecticstructures on exploded fibrations.
Definition 2.52. A symplectic exploded T fibration (B, ω) is an even dimen-sional exploded T fibration B with a two form ω so that each point has a neigh-borhood equal to an open set in R2n × Tm
A with a two form defined as follows
ω := ωR2n − i∑
j
dzαj ∧ dzα
j
Here ωR2n is the standard form on R2n and αj is some finite set generatingA∗.
The second part of this symplectic form can be written as
∑
i
d(αidθi)
αi =∑
j
αji
∣
∣
∣zα
j∣
∣
∣
2
or α :=∑
j
∣
∣
∣zα
j∣
∣
∣
2
αj
By dθi, we mean the imaginary part of z−1dz. The Hamiltonian torus actionon this standard model given by the vectorfields ∂
∂θihas moment map given by
α := (α1, . . . , αn) ⊂ Rn. The image of this moment map is the dual cone to Adefined by
α so that α · a ≥ 0 for all a ∈ A
Example 2.53.
Suppose that we have a compact symplectic manifold M with some collec-tion of embedded symplectic submanifolds Ni which intersect each other or-thogonally. (In other words, if x ∈ Ni ∩Nj , the vectors v ∈ TxM so that ω(v, ·)vanishes on TxNi are contained inside TxNj .) Then a standard Moser typeargument gives that there exists some collection of neighborhoods Ui of Ni sothat:
1. Ui is identified with some neighborhood of the zero section in a complexline bundle over Ni (with T action given by multiplication by eiθ), and aT invariant connection one form αi so that ω = ωNi
+ d(r2i αi), where riindicates some radial coordinate on the line bundle, Ni is identified withthe zero section, and ωNi
indicates the pullback of the form ω under theprojection to Ni.
2. These identifications are compatible in the sense that on⋂
i∈I Ui, all thedifferent projections to Ni and the T actions commute, and
ω = ωT
Ni+∑
d(r2i αi)
where ωT
Niis the pullback of ω under the composition of projection maps
to⋂
Ni.
28
We can associate an exploded T fibration to the above as follows:
1. Choose coordinate charts V on M which are identified with open balls inCn so that the intersection of V with the submanifolds Ni is equal to theunion of the planes zk = 0. If Ni intersects this coordinate chart, it doesso where some coordinate zki
= 0. Choose these charts so that the fibersof the complex line bundle over Ni are identified with the slices where allother coordinates are constant, and the T action is multiplication of zki
by eiθ, and ri = |zki|.
2. Replace V ⊂ Cn with the corresponding subset of Tnn, where each coordi-
nate zi is replaced by the standard coordinate zi. Replacing zi with zi intransition functions between these coordinates gives transition functionswhich are smooth exploded isomorphisms, which defines an exploded fi-bration M. A standard Moser type argument then gives that the two formon M corresponding to ω is in the form defined above.
The following is a picture of the result of this procedure on a toric symplecticmanifold, where for our symplectic submanifolds we use the fixed loci of circleactions. We shall represent the smooth part ⌈M⌉ by its moment map:
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0
⌈M⌉
[M]
⌊M⌋
Definition 2.54. An almost complex structure J on a symplectic exploded T
fibration (B, ω) is tamed by ω if ω(v, Jv) is positive for any vector v so thatvf 6= 0 for some smooth function f .
The above definition of taming is clearly inadequate for taming holomorphiccurves in Tn where it is satisfied trivially by a two form that is identically 0because all smooth functions on Tn are locally constant. We remedy this asfollows: (Note that we shall use freely the fact that any holomorphic curve inB lifts to a holomorphic curve (with a refined domain) in B′ for any refinementB′ −→ B.)
Definition 2.55. A strict taming of a complete almost complex exploded T
fibration B is a set Ω of closed two forms on refinements of B containing at leastone symplectic form. Each form in Ω must be nonnegative on J holomorphicplanes, and there must exist a metric g on B and a positive number c > 0smaller than the injectivity radius of g so that
29
1. Given any smooth exploded T curve f : C −→ B, the ω energy,∫
Cf∗ω is
a constant independent of ω ∈ Ω
2. Given any point p −→ B, there exists a taming form ωp ∈ Ω so that onthe ball of radius c around p,
ωp(v, Jv) ≥ g(v, v)
The Ω energy of a map f : C −→ B where C is two dimensional and orientedis given by
EΩ(f) := supω∈Ω
∫
f∗ω
For example, any compact almost complex manifold tamed by a symplecticform is strictly tamed by that symplectic form. A strict taming is what is neededto get local area bounds on holomorphic curves.
The following example is very important for applications of the theory ofholomorphic curves in exploded T fibrations.
Lemma 2.56. Suppose that a symplectic exploded fibration (B, ω) satisfies thefollowing:
1. B is basic and complete.
2. All strata of the base ⌊B⌋ have the integral affine structure of either astandard simplex or t[0,∞)n
If the above hold, then there exists some civilized almost complex structure Jand a strict taming Ω of (B, J) so that ω ∈ Ω.
Proof:Each strata Bi is in the form of some Tn
⌊Bi⌋bundle over the interior of a
smooth compact symplectic manifold (Mi, ωMi) with a collection of orthogo-
nally intersecting codimension 2 embedded symplectic submanifolds Mj like inexample 2.53 on page 28. (There is one such Mj for each strata ⌊Bj⌋ whichhas ⌊Bi⌋ as a boundary face). By interior we mean the complement of thesesymplectic submanifolds.
As in example 2.53, there exists a neighborhood of Mj , Ui,j ⊂ Mi which isidentified with a complex line bundle over Mi with radial coordinate ri,j andconnection one form αi,j so that ωMi
= ωMj+ d(r2i,jαi,j). We can make these
identifications compatible in the sense that if Mk is the intersection of Mj1 andMj2 , then on Ui,j1 ∩ Ui,j2 ,
ωMi= ωMk
+ d(r2i,j1αi,j1 ) + d(r2i,j2αi,j2)
the T actions and projections from the line bundles commute, and ri,j1 andαi,j1 are the pullback of rj2,k and αj2,k under the projection to Mj2 . (To con-struct these identifications compatibly, start with the strata ⌊Bi⌋ of highestdimension.)
30
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋ Mi := ⌈Bi⌉
Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1
ri,j2
rj1,k
rj2,k
We can arrange this so that in local coordinates in Ui,j the fiber coordinatez is actually the smooth part of some coordinate z from a coordinate chart onB. As B is complete, there exists some r0 > 0 so that the tubular neighborhoodaround Mj of radius r0 is always contained inside Ui,j .
We can choose an almost complex structure J tamed by ω so that on Ui,j ,J is the usual complex structure restricted to the fibers of our line bundle, andis determined by this and the lift (using our connection αi,j) of some almostcomplex structure Jj on Mj . (Note that such a J will be civilized in the senseof definition 2.36 on page 22.)
Denote by Uj a connected open subset of B which is the preimage of theunion of all Ui,j in ⌈B⌉. (This is an open subset that contains the strata Bj .)
There exist R∗tR+
valued functions ri,j defined on Uj who’s smooth part is equalto ri,j , so that locally there exist standard coordinates so that ri,j = ck |zk|,and so that
∏
i⌊ri,j⌋ = tc if ⌊Bj⌋ is simplex of finite size. Note that if ⌊Bj⌋ hasnonzero dimension, then any adjacent strataBk with ⌊Bk⌋ of nonzero dimensionhas the same size, and must obey a similar equation
∏⌊ri,k⌋ = tc with exactlythe same constant tc.
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k
⌊r1,j⌋
⌊r2,j⌋ ⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋
We now provide a strict taming Ω which for each point p −→ B includessome symplectic form ωp taming J well in a neighborhood of p as in part 2 ofthe above definition of a strict taming. We do this as follows: Suppose thatp ∈ Uj so that ri,j(p) < r0
2 and rj,i ≥ r02 for all i so that this makes sense.
Consider the set Sp ⊂ R∗tR given by the coordinates of p, ri,j(p). We shall
31
define ωp on a refinement of B determined by a subdivision of the base ⌊B⌋ asin example 2.6 on page 27. Subdivide the strata ⌊Bj⌋ by the set of all planesof the form ⌊ri,j⌋ ∈ ⌊Sp⌋. (This is the set of planes through p parallel tothe boundary of the strata and their image under any symmetry of the strata.)Similarly subdivide all strata Bk connected to Bj through strata with nonzerodimensional bases by the planes ⌊ri,j⌋ ∈ ⌊Sp⌋. This subdivision of ⌊B⌋ definesa refinement of B as in example 2.6.
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋
⌊p⌋
We can now define a symplectic form ωp on the refinement of B definedabove as follows: First choose a function
φ : R∗tR −→ R
so thatφ(r) = 0 ∀r > 2t0
φ(r) + φ(r−1) = −1and considered as a function φ : R∗ −→ R, φ is smooth, increasing and
dφ(r)
dr>
1
rfor all
1√2< r <
√2
Also choose some smooth monotone function ρ : R −→ R so that ρ(x) = x for
all x ≥ r20 and ρ(x) =r202 for all x <
r204 .
Now define h : R∗tR+ −→ R by
h(r) := ρ(⌈r2⌉) + r202 |Sp|
∑
x∈Sp
φ(rx−1)
We shall be replacing r2 with h(r) in what follows: the relevant properties ofh is that it agrees with r2 for r large enough, is 0 for r small enough, andhas sufficiently large derivative close to points where r ∈ Sp, and is monotoneincreasing. On Uj we have that ω = ωMj
+∑
d(r2i,jαi,j), we shall replace thiswith
ωp := ωMj+∑
d(h(ri,j)αi,j))
Now we do the same for all strata Bk connected to Bj through strata withbases of non zero dimension setting on Uk
ωp := ωMk+∑
d(h(ri,k)αi,k))
32
For all other Ui, we simply leave ω unchanged. This defines ωp as a two formon our refinement. It is now not difficult to check that the set of forms Ω givenby ω and ωp for all p is a strict taming of (B, J).
For example, the exploded fibration constructed in example 2.53 from a com-pact symplectic manifold with orthogonally intersecting codimension 2 symplec-tic submanifolds admits an almost complex structure with a strict taming. Thisis relevant for considering Gromov Witten invariants relative to these subman-ifolds.
The following picture of a symplectic form on a refinement in a toric casemay be helpful for understanding the construction in the above proof. (We havedrawn ⌈B⌉ in moment map coordiantes.)
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋
⌈B⌉ ⌊B⌋
⌈B′⌉ ⌊B′⌋
Example 2.57.
Given any symplectic B with a compact tropical part ⌊B⌋ satisfying the
conditions of Lemma 2.56, we can construct a family B −→ T11 of symplectic
exploded fibrations so that the fiber over z = 1t1 is B, and the fiber over z = 1is a smooth compact symplectic manifold.
2.8 Moduli stack of smooth exploded T curves
We shall use the concept of a stack without giving the general definition. (Seethe article [3] for a readable introduction to stacks). The reader unfamiliar withstacks may just think of our use of stacks as a way of keeping all informationabout families of holomorphic curves in case it is needed for future papers.
When we say that we shall consider an exploded T fibration B as a stack,we mean that we replace B with a category B over the category of explodedT fibrations (in other words a category B with a functor to the category ofexploded T fibrations) as follows: objects are maps into B:
A −→ B
33
and morphisms are commutative diagrams
A −→ B
↓ ↓ idC −→ B
The functor from B to the category of exploded T fibrations is given by sendingA −→ B to A, and the above morphism to A −→ C.
Note that a maps B −→ D are equivalent to functors B −→ D whichcommute with the functor down to the category of exploded fibrations. Suchfunctors are morphism of categories over the category of exploded fibrations,and this is the correct notion of maps of stacks.
For example, a point thought of as a stack is equal to the category of explodedfibrations itself, with the functor down to the category of exploded fibrationsthe identity. We shall refer to points thought of this way simply as points.
Definition 2.58. The moduli stackMsm(B) of smooth exploded T curves in B
is a category over the category of exploded fibrations with objects being familiesof smooth exploded curves consisting of the following
1. A exploded T fibration C
2. A pair of smooth exploded T morphisms
C −→ B
π ↓F
3. A section j of ker(dπ) ⊗ (T ∗C/π∗(T ∗F))
so that
1. π : C −→ F is a family (definition 2.45 on page 25).
2. The inverse image of any point p −→ F is an exploded T curve withcomplex structure j.
A morphism between families of curves is given by exploded morphisms fand c making the following diagram commute
F1 ←− C1 −→ B
↓ f ↓ c ↓ idF2 ←− C2 −→ B
so that c is a j preserving isomorphism on fibers.The functor down to the category of exploded fibrations is given by taking the
base F of a family.
Note that morphisms are not quite determined by the map f : F1 −→ F2.C1 is non-canonically isomorphic to the fiber product of C2 and F1 over F2.
This is a moduli stack in the sense that a morphism F −→ Msm(B) isequivalent to a family of smooth curves F←− C −→ B (this is the family whichis the image of the identity map F −→ F).
34
Definition 2.59. A holomorphic curve C −→ B is stable if it has a finite num-ber of automorphisms, and is not a nontrivial refinement of another holomorphiccurve. (If B is basic, this is equivalent to all smooth components of C which aremapped to a point in ⌈B⌉ being stable as punctured Riemann surfaces.)
Definition 2.60. A family of stable holomorphic curves in B is a family ofsmooth curves so that the map restricted to fibers is holomorphic and stable.The moduli stack of stable holomorphic curves in B, is the substack M(B) ⊂Msm(B) with objects consisting of all families of stable holomorphic curves,and morphisms the same as inMsm.
Definition 2.61. Given a closed 2 form ω on the exploded T fibration B, thesmooth exploded T curve f : C −→ B is ω-stable if the integral of f∗ω isnonnegative on any smooth component, and positive on any smooth componentwhich is unstable as a punctured Riemann surface.
Use the notation Msm,ω(B) to indicate the substack of Msm consisting offamilies of ω-stable smooth curves.
We have M(B) ⊂ Msm,ω(B) ⊂ Msm(B). The moduli stack of ω-stablecurves is a little better behaved than the moduli stack of stable curves becauseω-stable curves have only a finite number of automorphisms.
We want to make compactness statements about holomorphic curves in fam-ilies, so we generalize the above definitions for a family B −→ G as follows:
Definition 2.62. The moduli stack of smooth curves in a family B −→ G,Msm(B → G) is the substack of Msm(B) which is the full subcategory whichhas as objects families which admit commutative diagrams
(C, j) −→ B
↓ ↓F −→ G
The moduli stack of ω-stable smooth curvesMsm,ω(B→ G) and the moduli
stack of stable holomorphic curvesM(B → G) are defined as the appropriate
substacks ofMsm(B→ G). Note that there is a morphismMsm(B→ G) −→G which sends the object given by the diagram above to F −→ G. The appro-priate compactness theorem for families states that if we restrict to the part ofthe moduli space with bounded energy and genus, the mapM(B→ G) −→ G
is topologically proper.
The goal of this paper is to prove that the moduli stack of finite energy stableholomorphic curves with a fixed number of punctures and genus in a completeexploded T fibration with a strict taming is topologically compact. We shouldsay what this means in this context. First, we need a notion of topologicalconvergence. The following definition can be thought of as a notion of what itmeans for a sequence of points inMsm to converge topologically.
Definition 2.63. A sequence of points pi −→M(B) corresponding to smooth
curves f i : Ci −→ B converges topologically to f : C −→ B corresponding top −→M(B) in C∞,δ if there exist a sequence of families
35
F←− (C, ji)fi
−→ B
so that this sequence of families converges in C∞,δ to
F←− (C, j)f−→ B
and a sequence of maps pi of a point into F so that pi converges to p in ⌈F⌉,f i is the map given by the restriction of f i to the fiber over pi, and f is givenby the restriction of f to the fiber over p.
(It may seem strange that we use C∞,δ instead of smooth convergence forthe moduli stack of smooth curves. This is an artifact of our methods of proof.It is highly likely that in the case of integrable complex structures the analogoustheorem can be proven with smooth convergence replacing C∞,δ convergence.It is also perhaps more natural for our case to consider the moduli stack of C∞,δ
families. The required results in this setting follow trivially from results in thesmooth setting. It is expected thatM carries a natural type of C∞,δ Kuranishistructure.)
We say that a sequence of points in Msm(B → G) converge topologically
if they converge topologically in Msm(B). Topological convergence in M andMsm,ω is defined to simply be topological convergence inMsm. We say that asequence of points in G converges topologically if and only if the correspondingsequence of points in G converges topologically.
Definition 2.64. To say that M is topologically compact in C∞,δ means thatgiven any sequence of points in M, there exists a subsequence that convergestopologically in C∞,δ to some point in M.
To say that map of stacks f :M−→ G is topologically proper in C∞,δ is tosay that if pi −→M is some sequence of points so that f(pi) converges topolog-
ically in G, then there exists a subsequence of pi that converges topologically
in C∞,δ to some point in M.
In some sense, the moduli stack of stable holomorphic curves should becomplete. (The sense in which this is true involves dealing with transversalityissues.) We shall not say much about the other requirement for completeness inthis paper, but note that if U ⊂ T is a connected open subset of T, and U −→Msm,ω is a family of smooth curves so that at least one curve is holomorphic,then the family is holomorphic U −→M.
The main theorem of the paper, on page 54 can now be understood.
3 Estimates
In this section, we shall prove the analytic estimates required for our com-pactness theorem. These estimates do not differ much from the standard esti-mates used to study (pseudo)holomorphic curves in smooth symplectic mani-folds. They are also proved using roughly the same methods as in the smoothcase. We shall make the following assumptions:
36
1. We’ll assume that our target B is complete. This is required to give usthe kind of bounded geometry found in the smooth case when the targetis compact or has a cylindrical end. We shall also assume that B is basic.The assumption that B is basic is mainly for convenience in the argumentsthat follow. It is most useful in establishing coordinates and notation forthe construction of families of smooth curves in the proof of Theorem 4.1.
2. We’ll assume our almost complex structure J on B is civilized. Thisartificial (but easy to achieve) assumption means that J induces a smoothalmost complex structure on the smooth parts of B. This allows us touse standard estimates from the study of holomorphic curves in smoothmanifolds with minimal modification. This assumption is needed to haveholomorphic curves be smooth morphisms instead of just C∞,δ morphisms.
3. We’ll assume that J has a strict taming Ω. This is required to get a localarea bound for holomorphic curves. We shall also work with a particularsymplectic form ω ∈ Ω. This together with J defines a pseudo metricon B which collapses all tropical directions (so it is like a metric on thesmooth part ⌈B⌉.) The strategy of proof for most of our estimates isroughly as follows: first prove a weak estimate in ω’s pseudo metric, andthen improve this to get an estimate in an actual metric on B. We shalluse this strategy to prove estimates for the the derivative at the center ofholomorphic disks with bounded Ω energy and small ω energy, and to getstrong estimates on the behavior of holomorphic cylinders with boundedΩ energy and small ω energy.
This section ends with Proposition 3.17, which is a careful statement of thefact that any holomorphic curve with bounded Ω energy, genus, and numberof punctures can be decomposed into a bounded number of components, whicheither have ‘bounded conformal structure and bounded derivative’, or are annuliwith small ω energy, (and so we have strong estimates on their behavior.)
Throughout this section, we shall use the notation (B, J,Ω, ω, g) to indicatea smooth, basic, complete exploded fibration B, a civilized almost complexstructure J , a strict taming Ω, a choice of symplectic taming form ω ∈ Ω, anda metric g. If we say that a constant depends continuously on (B, J,Ω, ω, g),
we mean that if we have a family B −→ G with such a structure on each fiber,the constant can be chosen a continuous function on G. In what follows, wewill generally only prove lemmas for a single exploded fibration B, and leavethe proof of the corresponding statement for families to the diligent reader.
We shall use the following lemma which bounds the geometry of a completeexploded T fibration. Note that given a metric g on some complete B, theinjectivity radius is a continuous function on B, and is therefore bounded below.
Lemma 3.1. Given a complete exploded T fibration B with a metric g and anyfinite number of smooth tensor fields θi (such as an almost complex structureor a symplectic structure), for any R > 0 which is smaller than the injectivityradius of B, if a sequence of points pn −→ B converges topologically to p −→ B,then (BR(pn), g, θi) converges to (BR(p), g, θi) in the following sense:
Considering BR(pi) as a smooth manifold, there exist a sequence of diffeo-morphisms fn : BR(pn) −→ BR(p) so that f∗
n(g) and f∗n(θi) converges in C∞
on compact subsets of BR(pn) to g and θi.
37
Proof:This follows from the fact that all smooth tensor fields (and their derivatives)
can be locally written as some sum of smooth functions on B times some basistensor fields.
This tells us that the geometry of a complete exploded T fibration is boundedin the same way that the geometry of a compact manifold is bounded. Theanalogous result for families is also true.
The following is a consequence of our bounded geometry and the standardmonotonicity lemma for holomorphic curves first proved in [5]. (The proofinvolves a bound on the covariant derivatives of J , the curvature of g and theinjectivity radius, all of which vary continuously in families.)
Lemma 3.2. Given (B, J,Ω, g), for any ǫ > 0 there exists some E > 0 depend-ing continuously on (B, J,Ω, g) so that any non constant J holomorphic curvewhich passes through a point p and which is a complete map when restricted tothe ǫ ball around p has Ω energy greater than E.
Lemma 3.3. Given (B, J,Ω, ω), define the pseudo-metric
〈v, w〉ω :=1
2(ω(v, Jw) + ω(w, Jv))
For any such (B, J,Ω, ω) and a choice of ǫ > 0, there exists some E dependingcontinuously on (B, J,Ω, ω) so that any non constant J holomorphic curve whichpasses through a point p and which is a complete map when restricted to the ǫball around p in the above pseudo metric has ω energy greater than E.
Proof:The fact that B is basic and complete means in particular that it can be
covered by a finite number of coordinate charts, U ⊂ R2n×TmA . R2n×Tm
A withJ and ω can be regarded as subsets of more standard charts R2n × Tk
l cut outby setting some monomials equal to 1. This has the advantage that the smoothpart of this is just equal to R2n × Cl, and we can assume our symplectic andalmost complex structures just come from smooth ones here (this uses that Jis civilized), so we can use standard holomorphic curve results. Note also thatthe above pseudo metric is actually a metric on this smooth part. The size ofcovariant derivatives of J in this metric give continuous functions on B, andare therefore bounded because B is complete. To have the bounded geometryrequired to apply the standard monotonicity lemma for holomorphic curves, wealso need some kind of injectivity radius estimate. For a point q ∈ ⌈U⌉ in thesmooth part of a coordinate chart, we can define iU (q) to be the injectivityradius at q of ⌈U⌉ with our metric. We can then define an injectivity radius fora point p ∈ [B] to be given by
i(p) := infq
(
d(p, q) + maxU
iU (q))
where d(p, q) indicates distance in our pseudo metric. This defines a continuousfunction on B, and at each point i(p) > 0, so the fact that B is complete tellsus that there is some lower bound 0 < c ≤ i(p).
38
To prove our lemma we can now take 0 < ǫ < c. If our holomorphic curveis complete restricted to the ǫ ball around p, it is also complete (which impliesthat the maps of smooth components are proper maps) when restricted to theǫ ball around any q ∈ ⌈U⌉ so that d(p, q) = 0. We can apply the standardmonotonicity lemma for holomorphic curves to these balls, so we can choosesome E > 0 so that any holomorphic curve passing through q which is completerestricted to the ball around q is either constant in ⌈U⌉ or has energy greaterthan E. So either our holomorphic curve must have energy greater than E orit must have some connected component which is constant in the smooth partof every coordinate chart. This connected component must therefore not touchthe boundary of our ǫ ball, and must be a complete map to B. The fact that Ωis a strict taming then allows us to use Lemma 3.2 to complete our proof.
The above proof also implies the following:
Lemma 3.4. Given (B, J,Ω, ω), there exists some ǫ > 0 which can be chosento depend continuously on (B, J,Ω, ω) so that given any complete holomorphiccurve in B, the ω energy of any smooth component is either greater than ǫ, orthe image of that smooth component has zero size in the ω pseudo-metric.
Lemma 3.5. Given (B, J,Ω, ω), for any ǫ > 0, there exists some E > 0depending continuously on (B, J,Ω, ω) so that any J holomorphic map f ofe−(R+1) < |z| < e(R+1) with ω-energy less than E is contained inside a ball ofradius ǫ (in ω’s pseudo metric) on the smaller annulus e−R ≤ |z| ≤ eR.
Proof:Suppose to the contrary that this lemma is false.Choose E small enough so that Lemma 3.3 holds for ǫ
8 balls. There must
be a path in e−(R+1) < |z| < eR+1 joining z = 1 with an end of this annuluscontained entirely inside the ball B ǫ
8(f(1)). (Otherwise, f restricted to some
subset would be a proper map to B ǫ8(f(1)), and we could use Lemma 3.3 to get
a contradiction.) Suppose without loss of generality that it connects 1 with thecircle |z| = e−(R+1).
Suppose for a second that there exists some point z so that e−(R+ 34 ) < |z| <
e−(R+ 14 ) and f(z) /∈ B 3ǫ
8(f(1)). As above, there must exist some path connect-
ing z with an end which is contained in the ball B ǫ8(f(1)). There must therefore
be some region conformal to [0, π]× [0, 14 ] so that the dist (f(0, t), f(π, t)) > ǫ
8 .Then the Cauchy-Schwartz inequality tells us that
∫ π
0
∫ 14
0
|df |2ω >1
4
(
ǫ8
)2
π
This in turn gives a uniform lower bound for the energy of our curve. This casecan therefore be discarded, and we can assume that all z satisfying e−(R+ 3
4 ) <|z| < e−(R+ 1
4 ) are contained in B 3ǫ8(f(1)).
As we are assuming this lemma is false, there must be some point z2 so thate−R ≤ |z2| ≤ eR and f(z2) /∈ Bǫ(f(1)). Repeating the above argument (withz2 in place of the point z = 1) gives us that z must be contained in B 3ǫ
8(f(z2))
when eR+ 14 ≤ |z| ≤ eR+ 3
4 .
39
We then have f contained in balls which are at least ǫ4 apart on the boundary
of an annulus, thus we can apply Lemma 3.3 to some point in the image under fof the interior at least ǫ
8 from both balls to obtain a lower bound for the energyand a contradiction to the assumption that the lemma was false.
The following gives useful coordinates for the analysis of holomorphic curves:
Lemma 3.6. Given (B, J,Ω, ω), there exist constants ci and ǫ > 0 dependingcontinuously on (B, J,Ω, ω) so that for all points p −→ B, there exists somecoordinate neighborhood U of p so that the smooth part, ⌈U⌉ can be identifiedwith a relatively closed subset of the open ǫ ball in Cn with almost complexstructure J and some flat J preserving connection ∇ so that
1. p is sent to 0, and the restriction of J to ⌈U⌉ is J .
2. The metric 〈·, ·〉ω on ⌈U⌉ is close to the standard flat metric on Cn so that
1
2〈v, v〉ω ≤ |v|2 ≤ 2〈v, v〉ω
3. J at the point z ∈ Cn converges to the standard complex structure J0 onCn as z → 0 in the sense that
∥
∥
∥J − J0
∥
∥
∥≤ c0 |z|
4. The torsion tensor
T∇(v, w) := ∇vw −∇wv − [v, w]
is bounded by c0, and has its kth derivatives bounded by ck.
The only point in this lemma which does not follow from the definition ofa civilized almost complex structure and calculation in local coordinates is thefact that the constants involved do not have to depend on the point p. Thisfollows from the fact that B is complete.
Lemma 3.7. Given a holomorphic map f of the unit disk to a space with a flat Jpreserving connection ∇, fx defines a map to Cn defined by parallel transportingfx back to the tangent space at f(0) which we then identify with Cn. Such anfx satisfies the following equation involving the torsion tensor of ∇, T∇.
∂fx =1
2JT∇(fy, fx)
Proof:
∂fx =1
2(∇fxfx + J∇fyfx)
=1
2∇fx(fx + Jfy) +
1
2J(∇fyfx −∇fxfy)
=1
2JT∇(fy, fx)
40
As the ∂ above is the standard linear ∂ operator, this expression is goodfor applying the following standard elliptic regularity lemma for the linear ∂equation. This allows us to get bounds on higher derivatives from bounds onthe first derivative of holomorphic functions.
Lemma 3.8. For for a given number k and 1 < p <∞, there exists a constantc so that given any map f from the unit disk D(1),
‖f‖Lp
k+1(D( 12 ))≤ c
(
∥
∥∂f∥
∥
Lp
k(D(1))
+ ‖f‖Lp
k(D(1))
)
Lemma 3.9. Given (B, J,Ω, ω), there exists some energy E, some distancer > 0, and a constant c, each depending continuously on (B, J,Ω, ω) so thatany holomorphic map f of a disk |z| ≤ 1 into B with ω-energy less than Eor contained inside a ball of ω radius r, satisfies
|df |ω < c at z = 0
We shall omit the proof of the above Lemma which is a standard bubblingargument, similar, but easier than the proof of the following:
Lemma 3.10. Given (B, J,Ω, ω, g), for any E > 0, there exists some ǫ > 0,distance r > 0 and constant c, each depending continuously on (B, J,Ω, ω, g) sothat any holomorphic map f of a disk |z| ≤ 1 into B with Ω-energy less thanE, and either contained in a ω-ball of radius r or having ω-energy less than ǫsatisfies
|df |g < c at z = 0
Proof:Suppose that this lemma was false. Then there would exist some sequence
of maps fi satisfying the above conditions with |dfi(0)| → ∞. First, we obtaina sequence of rescaled J holomorphic maps fi : D(Ri) −→ B from the standardcomplex disk of radius Ri so that
∣
∣
∣dfi
∣
∣
∣≤ 2
∣
∣
∣dfi(0)
∣
∣
∣= 1
limi→∞
Ri =∞
We achieve this in the same way as in any standard bubbling off argumentsuch as the proof of lemma 5.11 in [1]. We can then use Lemma 3.6, Lemma3.7, and Lemma 3.8 to get a bound on the higher derivatives of fi on D(Ri−1).
As B is complete, we can choose a subsequence so that fi(0) convergestopologically to some p −→ B. Lemma 3.1 tells us that the geometry of (g, J, ω)around fi(0) converges to that around p. Considering all our maps as mapssending 0 to p, we can choose a subsequence that converges on compact subsetsto a non constant holomorphic map f : C −→ B. Note that f either has ω-energy less than ǫ or is contained in a ball of radius r in the ω pseudo-metric.Because f must have finite ω energy, we can use Lemma 3.5 to tell us that fmust converge in the ω pseudo-metric as |z| → ∞. Then by choosing ǫ or r smallenough, we can prove using Lemma 3.3 and the standard removable singularity
41
theorem for holomorphic curves, that f must actually have zero size in the ωpseudo-metric because f either has ω energy less than ǫ or is contained in an ωball of radius r.
Now that we have that the image of f has zero size in the ω pseudo metric,we know that f must be contained entirely within some (C∗)n worth of pointsover a single topological point in [B]. This (C∗)n has the standard complexstructure, so f gives us n entire holomorphic maps from C −→ C∗. As f is nonconstant, at least one of these maps must be non constant. This map must haveinfinite degree, as it must have dense image in the universal cover of C∗ whichis just the usual complex plane. It therefore has infinite Ω-energy, contradictingthe fact that it must have Ω-energy less than E.
To prove the family case, it is important to note that (as stated at the start ofthis section), when we say our constants depend continuously on (B, J,Ω, ω, g),
we mean that given any finite dimensional family, (B −→ G, J,Ω, ω, g), theconstants can be chosen to depend continuously on G. With this understood,the proof in the family case is analogous to the above proof.
Lemma 3.11. Given (B, J,Ω, ω), for any 0 ≤ δ < 1, there exists an energybound E > 0, a distance r > 0, and a constant c, each depending continuouslyon (B, J,Ω, ω) so that any holomorphic map f from the annulus e−(R+1) < |z| <e(R+1) to B with ω energy less than E, or contained inside a ball of ω radius rsatisfies the following inequality
dist(f(z), f(1))ω ≤ ce−δR(|z|δ + |z|−δ) for e−R ≤ |z| ≤ eR
Actually, with a more careful argument, this is true with exponent δ = 1,but we will only prove the easier case.Proof:
For this proof we shall use coordinates z = et+iθ, and the cylindrical metricin which ∂
∂t, ∂∂θ are an orthonormal frame.
We choose our energy bound E > 0 small enough that we can use thederivative bound from Lemma 3.9, and Lemma 3.5 implies that the smallerannulus is contained a small enough ball that we can use the coordinates ofLemma 3.6 (we also choose r small enough that this is true). We then have thefollowing estimate for the (standard) ∂ of f in these coordinates.
∣
∣∂f∣
∣ ≤ c1 |df | |f |We can run this through the inequality from Lemma 3.8 to obtain the fol-
lowing estimate on ∂f restricted to the interior of a disk (which holds on theinterior of the cylinder where we have the derivative bound from Lemma 3.9.)∥
∥∂f∥
∥
Lp(D( 12 ))≤ c1 ‖f‖Lp
1D( 12 )‖f‖L∞(D( 1
2 ))≤ c2 ‖f‖L∞(D(1)) ‖f‖L∞(D( 1
2 ))(1)
Here c2 is a constant depending only on p and (B, J, ω). We can fix p to besomething bigger than 2. By choosing E or r small, we can force |f | to be assmall as we like on the smaller cylinder using Lemma 3.5.
Now we can use Cauchy’s integral formula
2πif(z0) = −∫
|z|=1
f(z)
z − z0dz +
∫
|z|=e2l
f(z)
z − z0dz +
∫
1≤|z|≤e2l
∂f(z)
z − z0∧ dz
42
or in our coordinates,
f(t0, θ0) = −1
2π
∫ 2π
0
f(0, θ)
1− et0ei(θ−θ0)dθ +
1
2π
∫ 2π
0
f(2l, θ)
1− et0−2lei(θ−θ0)dθ
+1
2π
∫ 2π
0
∫ 2l
0
∂f(θ, t)
1− et0−tei(θ−θ0)dtdθ
Let us now consider each term of this expression for f(l, θ0) in the middle ofa cylinder under the assumption that the average of f(2l, θ) is 0.
The first term:∣
∣
∣
∣
1
2π
∫ 2π
0
f(0, θ)
1− elei(θ−θ0)dθ
∣
∣
∣
∣
≤ 1
el − 1max |f(0, θ)|
The second term using that the average of f(2l, θ) is 0,
∣
∣
∣
∣
1
2π
∫ 2π
0
f(2l, θ)
1− el−2lei(θ−θ0)dθ
∣
∣
∣
∣
=
∣
∣
∣
∣
1
2π
∫ 2π
0
f(2l, θ)
(
1
1− e−lei(θ−θ0)− 1
)
dθ
∣
∣
∣
∣
=
∣
∣
∣
∣
1
2π
∫ 2π
0
f(2l, θ)
(
e−lei(θ−θ0)
1− e−lei(θ−θ0)
)
dθ
∣
∣
∣
∣
≤ 1
el − 1max |f(2l, θ)|
The third term:∣
∣
∣
∣
∣
1
2π
∫ 2π
0
∫ 2l
0
∂f(θ, t)
1− et0−tei(θ−θ0)dtdθ
∣
∣
∣
∣
∣
≤∥
∥∂f∥
∥
L3
1
2π
∥
∥
∥
∥
1
1− el−tei(θ−θ0)
∥
∥
∥
∥
L32
≤ c3(l + 1)
(
maxt∈[−1,2l+1]
|f(t, θ)|)(
maxt∈[0,2l]
|f(t, θ)|)
The constant c3 depends only on (B, J, ω). It is zero if J is integrable.To summarize the above, we have the following expression which holds if the
average of f(θ, 2l) is 0. (It also holds if the average of f(θ, 0) is 0 due to thesymmetry of the cylinder.)
|f(l, θ)| ≤ 1
el − 1(max |f(0, θ)|+max |f(2l, θ)|)
+ c3(l + 1)
(
maxt∈[−1,2l+1]
|f(t, θ)|)(
maxt∈[0,2l]
|f(t, θ)|) (2)
The amount that the average of f(t0, θ) changes with t0 inside [0, 2l] isdetermined by the integral of ∂f , which is dominated as above by a term of theform
change in average ≤ c4l
(
maxt∈[−1,2l+1]
|f(t, θ)|)(
maxt∈[0,2l]
|f(t, θ)|)
(3)
Now let’s put these estimates into slightly more invariant terms: Define thevariation of f(t, θ) on [a, b]× T1 as follows:
V f([a, b]) := maxt1,t2∈[a,b]
dist(f(t1, θ1), f(t2, θ2))ω
43
Now the above two estimates give the following:
V f([a, a+l]) ≤(
8
el − 1+ c5(l + 1)V f([a− l − 1, a+ 2l+ 1])
)
V f([a−l, a+2l])
Now if we choose l large enough, and then make V f small enough by makingour energy bound E small (or directly making r small), we can get the estimate
V f([a, a+ l]) ≤ e−δl
3V f([a− l, a+ 2l]) for [a− l − 1, a+ 2l+ 1] ⊂ [−R,R]
Applying this 3 times, we get that on the appropriate part of the cylinder,
V f([a− l, a+ 2l]) ≤ e−δlV f([a− 2l, a+ 3l])
so V f([a, a+ l]) ≤ e−δ2l
3V f([a− 2l, a+ 3l])
Inductively continuing this argument gives that if [a − nl, a + (n + 1)l] ⊂[−R+ 1, R− 1],
V f([a, a+ l]) ≤ e−δnl
3V f([a− nl, a+ (n+ 1)l])
The required estimate follows from this.
Lemma 3.8, Lemma 3.7, and the coordinates from Lemma 3.6 from page 40can be used with inequality 1 on page 42 in a standard fashion to get a similarestimate on the derivative of f in the ω pseudo metric. This gives the followingcorollary:
Corollary 3.12. Given (B, J,Ω, ω), there exists some energy E > 0 dependingcontinuously on (B, J,Ω, ω) so that given any c > 0, there exists some distanceR depending continuously on (B, J,Ω, ω, c) so that given any holomorphic mapf of a cylinder e−l−R < |z| < el+R to B with ω energy less than E, the ω energyof f restricted to e−l < |z| < el is less than cE.
Lemma 3.13. Given (B, J,Ω, g), and any energy bound E <∞ and exponent0 ≤ δ < 1, there exists a covering of B by a finite number of coordinate chartsand a constant c so that the following is true:
Given any holomorphic map f of Ω energy less than E from a cylindere−(R+1) ≤ |z| ≤ e(R+1) to B contained inside a coordinate chart, there exists amap F given in coordinates as
F (z) := (c1zα1ta1 , . . . , ckz
αktak , ck+1, . . . , cn)
so that
dist (f(z)− F (z)) ≤ ce−δR(
|z|δ + |z|−δ)
for e−R ≤ |z| ≤ eR
where for i ∈ [1, k], ci ∈ C∗, ai ∈ R, αi ∈ Z, and for i ∈ [k + 1, n], ci ∈ R. distindicates distance in the metric g.
In the case that we a have a family (B −→ G, J,Ω, g) with the above struc-
ture, we can choose c continuous on G and a set of coordinate charts U on B
so that the above coordinate charts on any fiber are the non empty intersectionsof these with the fiber.
44
Proof:First, choose any symplectic form ω ∈ Ω. Because B is basic and complete,
we can then choose a finite number of coordinate charts covering B which aresmall enough in the ω pseudo metric so that Lemma 3.11 tells us that forany holomorphic map from a cylinder as above contained inside one of thesecoordinate charts satisfies
distω(f(z), f(1)) ≤ ce−δ′R(|z|δ′
+ |z|−δ′
) (4)
where δ′ = δ+12 .
In our coordinates,
f(1) = (c1ta1 , . . . , ckt
ak , ck+1, . . . , cn)
We can choose αi so that the winding numbers of the first k coordinates off(eiθ) are the same as our model map F :
F := (c1zα1ta1 , . . . , ckz
αk tak , ck+1, . . . , cn)
The metric g can be compared to the pseudo metric from ω on the lastcoordinates ck+1, . . . , cn, so we only need to prove convergence in the first kcoordinates. Let us do this for f restricted to the first coordinate f1.
For this, use coordinates z = et+iθ on the domain with the usual cylindricalmetric and the similar cylindrical metric on our target. (Choose our coordinatecharts so that this metric on the target is comparable to g). Use the notation
f1(z)
c1zα1ta1= eh(z)
where h is a C valued function so that h(0) = 0. The metric we are using onthe target is the standard Euclidean metric on C. Our goal is to prove theappropriate estimate for h.
First note that Lemma 3.8 and Lemma 3.7 can be used with the inequality4 in a standard fashion to get a similar estimate on the derivative of f in the ωpseudo metric.
|df(z)|ω ≤ ce−δ′R(|z|δ′
+ |z|−δ′) for e−R ≤ |z| ≤ eR (5)
(The c in the above inequality is some new constant which is independent off). We can choose our coordinate charts so that if J0 indicates the standardcomplex structure, there exists some constant M so that
|Jv − J0v|g ≤M |v|ωThis follows from writing down J in coordinates, and the fact that our ω
pseudo metric is a metric on the smooth part of our coordinate chart. This thentells us that
∣
∣∂f1∣
∣
gin our coordinates is controlled by |df |ω. This then gives
that for some new constant c independent of f we get the following inequality:
∣
∣∂h(z)∣
∣ ≤ ce−δ′R(|z|δ′
+ |z|−δ′
) for e−R ≤ |z| ≤ eR (6)
If we choose our coordinate charts small enough in the ω pseudo-metric,Lemma 3.10 gives us a bound for |df |g on the interior of the cylinder, so wehave that there exists some c independent of f so that
45
|dh(z)| < c for e−R ≤ |z| ≤ eR (7)
We can now proceed roughly as we did in the proof of Lemma 3.11. Inparticular, Cauchy’s integral theorem tells us that
h(t0, θ0) = −1
2π
∫ 2π
0
h(t0 − l, θ)
1− elei(θ−θ0)dθ +
1
2π
∫ 2π
0
h(t0 + l, θ)
1− e−lei(θ−θ0)dθ
+1
2π
∫ 2π
0
∫ t0+l
t0−l
∂h(t, θ)
1− et0−tei(θ−θ0)dtdθ
We can now bound each term in the above expression as in the proof ofLemma 3.11, except we use the estimate 6 to bound
∣
∣∂h∣
∣.Then we have the following estimate:
∣
∣
∣
∣
h(t0, θ0)−1
2π
∫ 2π
0
h(t0 + l, θ)dθ
∣
∣
∣
∣
≤ 1
el − 1max
∣
∣
∣
∣
h(t0 − l, θ)− 1
2π
∫ 2π
0
h(t0 + l, θ)dθ
∣
∣
∣
∣
+1
el − lmax
∣
∣
∣
∣
h(t+ l, θ)− 1
2π
∫ 2π
0
h(t0 + l, θ)dθ
∣
∣
∣
∣
+ c(l + 1)e−δ′Reδ′l(
eδ′t0 + e−δ′t0
)
(Of course, this is a new constant c, which is independent of l or h.)Note that the change in the average of h is determined by the integral of ∂h,
which we can bound using estimate 6. The change in this average can then beabsorbed into the last term of the above inequality. Define the variation of hfor a particular t as follows:
V h(t) := maxθ
∣
∣
∣
∣
h(t, θ)− 1
2π
∫ 2π
0
h(t, θ)dθ
∣
∣
∣
∣
We then have the following estimate (with a new constant c):
V h(t) ≤ 1
el − 1(V h(t+ l) + V h(t− l)) + c(l + 1)eδ
′(l−R)(
eδ′t + e−δ′t
)
Recalling that δ < δ′ and V h is bounded by equation 7, we can choose l largeenough so that for R sufficiently large, using the above estimate recursively tellsus that there exists some c independent of f or R so that
V h(t) ≤ ce−δR(eδt + e−δt) for −R ≤ t ≤ R
(This estimate follows automatically from the bound on V h for R bounded.)The required estimate for h then follows from the fact that the change in theaverage of h is bounded by the estimate 6, which is stronger than what we need(which is the same equation with δ in place of δ′). We therefore have
h(z) ≤ ce−δR(|z|δ + |z|−δ) for e−R ≤ |z| ≤ eR
which is the required estimate.
46
To prove compactness results, we shall divide our domain up into annuliwith small energy, and other compact pieces with derivative bounds. For thiswe shall need some facts about annuli. Recall the following standard definitionfor the conformal modulus of a Riemann surface which is an annulus:
Definition 3.14. The conformal modulus of an annulus A is defined as follows.Let S(A) denote the set of all continuous functions with L2 integrable derivativeson A which approach 1 at one boundary of A and 0 at the other. Then theconformal modulus of A is defined as
R(A) := supf∈S(A)
2π∫
A(df j) ∧ df
= supf∈S(A)
2π∫
A|df |2
We can extend the definition of conformal modulus to include ‘long’ annuliinside exploded curves as follows:
Definition 3.15. Call a (non complete) exploded curve A an exploded annulusof conformal modulus log xt−l if it is connected, and there exists an injectiveholomorphic map f : A −→ T with image 1 < |z| < xt−l. Call it an explodedannulus with semi infinite conformal modulus if it is equal to (a refinement of)|z| < 1 ⊂ T1
1.
(We use xt−l in the above definition because the tropical part of the resultingannulus will have length l.) Two exploded annuli with the same conformalmodulus may not be isomorphic, but they will have a common refinement.
We shall need the following lemma containing some useful properties of the(usual) conformal modulus.
Lemma 3.16. 1. An open annulus A is conformally equivalent to 1 < |z| <eR if and only if the conformal modulus of A is R. If the conformal mod-ulus of A is infinite, then A is conformally equivalent to either a punctureddisk, or a twice punctured sphere.
2. If Ai is a set of disjoint annuli Ai ⊂ A none of which bound a disk inA, then
R(A) ≥∑
R(Ai)
3. If A1 and A2 are annuli contained inside the same Riemann surface whichshare a boundary so that R(A2) < ∞, and the other boundary of A1
intersects the other boundary of A2 and the circle at the center of A2,then
R(A2) +16π2
R(A2)≥ R(A1)
(This is not sharp.)
47
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
A1 ∩ A2
A2
4. If A1 and A2 are annuli contained inside a Riemann surface so that everycircle homotopic to the boundary inside A2 contains a segment inside A1
that intersects both boundaries of A1, then
R(A1) ≤4π2
R(A2)
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
A1 ∩ A2
A2
0
1
Proof:The first two items are well known. To prove item 3, first set R = R(A2)
and put coordinates (0, R)× R/2πZ on A2. Consider any function f ∈ S(A1).Without losing generality, we can assume that the shared boundary is (0, θ),and that some segment of the other boundary of A1 is a curve in A2 between(R2 , 0) and (R, θ0) where θ0 ≥ 0. Then consider integrating |df |2 along diagonallines (Rt,−4πt+ c). Each of these lines contains a segment inside A1 on which
the integral of df is 1. The length of each segment is bounded by ((4π)2+R2)12 .
The integral of |df |2 along this segment is therefore at least (16π2+R2)−12 . This
tells us that the integral of |df |2 over A1 is at least 2πR((4π)2+R2) , and therefore,
R(A1) ≤(16π2 +R2)
R= R(A2) +
16π2
R(A2)
To prove item 4, consider a function f ∈ S(A1). We shall integrate |df |2along segments of the form (c, t) inside A2 ∩A1 traveling from one boundary ofA1 to the other. The integral of df along such a segment is 1, and the length ofthe segment is at most 2π, so the integral of |df |2 along the segment is at least12π , and the integral of |df |2 over A1 is at least R(A2)
2π . Therefore, we have
R(A1) ≤4π2
R(A2)
48
Proposition 3.17. Given (B, J,Ω, ω, g), an energy bound E and a numberN , and small enough ǫ > 0, there exists a number bound M depending lowersemicontinuously on (B, J,Ω, ω, g, E,N, ǫ) and for any large enough distanceR, a derivative bound c and conformal bound R depending continuously on(B, J,Ω, ω, g, E,N, ǫ, R) so that the following is true:
Given any complete, stable holomorphic curve f : C −→ B with energy atmost E, and with genus and number of punctures at most N , there exists somecollection of at most M exploded annuli Ai ⊂ C, so that:
1. Each Ai has conformal modulus larger than 2R
2. Put the standard cylindrical metric on Ai, and use the notation Al,i to de-note the annulus consisting of all points in Ai with distance to the boundaryat least l.
AR2 ,i ∩ AR
2 ,j = ∅ if i 6= j
3. f restricted to Ai has energy less than ǫ.
4. Each component of C−⋃AR,i is a smooth Riemann surface with boundedconformal geometry in the sense that any annulus inside one of thesesmooth components with conformal modulus greater than R must bounda smooth disk inside that component.
5. The following metrics can be put on Ai and each smooth component ofC−⋃AR,i:
(a) On any component which is a smooth torus, use the unique flat metricin the conformal class of the complex structure so that the area of thetorus is 1.
(b) If the component is equal to a disk, an identification with the standardunit disk can be chosen so that if AR,i is the bounding annulus, 0 isin the complement of Ai. Give components such as this the standardEuclidean metric.
(c) If the component is equal to some annulus, give it the standard cylin-drical metric on R/Z × (0, l). Give each Ai the analogous standardmetric.
(d) Any component not equal to a torus, annulus, or disk will admit aunique metric in the correct conformal class with curvature −1 sothat boundary components are geodesic (there will be no componentswhich are smooth spheres). Give these components this metric.
On any component of C−⋃A 9R10 ,i, the derivative in this metric is bounded
by c
|df |g < c
Moreover, on A 6R10 ,i − A 9R
10 ,i, the ratio between the two metrics defined
above is less than c.
6. There exists some lower energy bound ǫ0 > 0 depending only on ǫ, R, andE so that f restricted to any component of C − ⋃AR,i which is a disk,annulus or torus has ω energy greater than ǫ0
49
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
derivative bounded here
components with ω energy bounded below
A
AR:well behaved annulus
Proof:Lemma 3.10 tells us that for ǫ small enough, any holomorphic map of the unit
disk with ω-energy less than ǫ and Ω-energy less than E must have derivativeat 0 bounded by c0. We shall prove our theorem for ǫ small enough so that thisis true, and small enough that Corollary 3.12 also holds with an energy boundof ǫ. We shall also choose our distance R greater than 4π (for use with Lemma3.16), and large enough that Corollary 3.12 tells us that if f restricted to somesmooth annulus Ai has ω-energy less than ǫ, then the ω-energy of AR
2 ,i is less
than ǫ5 .
Let us now begin to construct our annuli. First, note that any edge of C haszero ω-energy, so we can choose an exploded annulus Ai containing each edgewith ω-energy less than ǫ
5 . (Note for use with item 6 that in the case that thisbounds a disk, the ω-energy of the resulting disk will be at least 4ǫ
5 ). We can dothis so that these Ai are mutually disjoint. Note that the complement of theseAi is a smooth Riemann surface with boundary. (Note also that no componentof C can be isomorphic to T as the Ω-energy of any complete component isequal to the ω-energy which must be zero on T. Similarly, there is no completecomponent of C which is locally modeled everywhere on T.)
For each connected component of C which is a smooth sphere, we shallnow remove an annulus. First, note that we can put some round metric in thecorrect conformal class of the sphere so that there exist 3 mutually perpendiculargeodesics which divide our sphere into 8 regions each of which has equal ω-energy. As our sphere must have energy at least ǫ in order to be stable, we canchoose two antipodal regions that each have energy ǫ
8 . By choosing ǫ0 > 0 smallenough, we can then get that there exist two disks with ω-energy at least ǫ0which intersect each of these regions, and which have radius as small as we like.
50
Choose ǫ0 > 0 small enough that the complement of these disks is an annulusof conformal radius at least k(2R+ 1) for some integer k > 5E
ǫ. How small ǫ0 is
required to achieve this depends only on R, ǫ and E. Then we can divide thisannulus into k annuli with conformal modulus at least (2R+1), at least one ofwhich has ω-energy at most ǫ
5 . Add this annulus to our collection. Note that itbounds disks which have energy at least ǫ0.
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
derivative bounded here
components with ω energy bounded below
AAR:well behaved annulus
ω-energy < ǫ5 A ω-energy > ǫ0
Consider a holomorphic injection of the unit disk i : D −→ C into thecomplement of our annuli AR,i constructed up to this point. Suppose that i(0)is in the complement of AR
4 ,i. Then Lemma 3.16 tells us that the restriction of
i to |z| < e−16π2
R is in the complement of AR2 ,i. Now suppose that |d(f i)| >
c0e16π2
R+k(2R+1) where k is some integer greater than 5E
ǫ. Then the restriction
of f to the disk |z| < e−16π2
R−k(2R+1) must have energy greater than ǫ due to
Lemma 3.10. This disk is surrounded by k disjoint annuli of conformal modulus(2R+ 1) which are contained in the complement of AR
2 ,i. At least one of these
must have energy less than ǫ5 . Add this annulus to our collection. Continue
adding annuli in this manner. After a finite (but not universally bounded)number of times, no more annuli can be added in this way. We shall argue thisbelow after adding some more annuli.
51
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
derivative bounded here
components with ω energy bounded below
AAR:well behaved annulus
ω-energy < ǫ5
Aω-energy > ǫ0
ARAR
2
New low energy annulus
Disk with ω-energy > ǫ0
We now want to add annuli so that condition 4 is satisfied. Suppose somesmooth component of C−⋃AR,i contains some annulus of conformal modulusgreater than 2R(3+ 5E
ǫ) that does not bound a disk. Note that as we have chosen
R > 4π, the size of such an annulus is greater than 2R(1 + 5Eǫ) + 2R + 32π2
R.
The annulus of size R at its end can’t be contained entirely within Ai, and then
applying Lemma 3.16 part 4 with the next annulus of size 16π2
Rin the place
of A1, we see that our annulus minus annuli at each end of conformal modulus
R+ 16π2
Rmust be contained entirely in the complement of AR
2 ,i. We then obtain
more than 5Eǫ
disjoint annuli in the compliment of AR2 ,i with conformal modulus
greater than 2R. The restriction of f to at least one of these annuli must haveω-energy less than ǫ. Add this annulus to our collection.
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
derivative bounded here
components with ω energy bounded below
AAR:well behaved annulus
ω-energy < ǫ5
Aω-energy > ǫ0
ARAR
2
New low energy annulusDisk with ω-energy > ǫ0
New low energy annulus
AR2
AR
We shall now argue that we can only add a finite number of annuli in thismanner. The number of annuli that bound disks is bounded by E
ǫplus the twice
the number of connected components of C that are spheres (which is boundedby E
ǫ). There are also a finite number of exploded annuli with infinite conformal
modulus. The number of these is bounded by our topological bounds, and thefact that there are at most E
ǫspherical components with fewer than 3 punctures.
Call the complement of all the above annuli C0. There is then a bound on thenumber of homotopy classes in C0 of our remaining annuli, given by our bounds
52
on the topology of C, the fact that we have removed a bounded number of annuli,and the observation that for any Riemann surface with boundary, there are onlya finite number of homotopy classes which contain annuli of conformal modulusgreater than 4π (Lemma 3.16 part 4 can help to prove this). If there were aninfinite number of annuli AR
2 ,i in our collection in the same homotopy class,
then there would exist an annulus in C0 in that homotopy class which containsall of them. As the AR
2 ,i are disjoint and have conformal modulus at least R,
this annulus would have to have infinite conformal modulus. Note that thereare no nontrivial annuli of infinite conformal modulus in C0 (because we haveremoved the annuli containing edges and at least one annulus from each sphericalcomponent), and if an annulus of infinite conformal modulus surrounds a diskwith energy greater than ǫ0, then the derivative of f must have been unboundedthere. This is not possible, so we must only have a finite number of annuli inour collection. Note that everything apart from our bound on the number ofannuli in a fixed homotopy class is bounded independent of f .
Now we shall merge some annuli so that there exist no annular component ofC−⋃AR,i with ω-energy less than ǫ
5 . Then we will have a bound on the numberof annuli which is independent of f . Suppose that we have some collectionA1, . . . ,An of our annuli so that AR,i and AR,i+1 bound an annulus whichhas ω-energy less than ǫ
5 . Use the notation A[m,n] to denote the annulus thatconsists of Am, An and everything in between. Then, as we’ve chosen each ofour Ai to have ω-energy less than ǫ
5 , A[i,i+2] has ω-energy less than ǫ, so we canapply Corollary 3.12 to show that far enough into the interior of this cylinder,there is very little ω-energy. (Note that as the parts of exploded annuli whichare locally modeled on T always have no ω-energy, so there is no difficulty inapplying this lemma in the seemingly more general setting of exploded annuli.)Applying Lemma 3.16 part 3, and noting that as we have chosen R > 4π, we
have 2R > R + 16π2
R, we see that A[i,i+2] − Ai − Ai+2 ⊂ AR
2 ,[i,i+2]. We have
chosen R large enough that Corollary 3.12 tells us that the energy of f restrictedto AR
2 ,[i,i+2] is less than ǫ5 . We can now repeat this argument inductively to
show that the energy of f restricted to A[1,n] is less than ǫ and AR2 ,[1,n] is less
than ǫ5 .
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
derivative bounded here
components with ω energy bounded below
AAR:well behaved annulus
ω-energy < ǫ5
Aω-energy > ǫ0
ARAR
2
New low energy annulusDisk with ω-energy > ǫ0New low energy annulus
AR2
ARA1
A2 A3A4
AR2 ,[1,3] has low ω-energy
Now, replace all of our sets of annuli A1, . . . ,An of maximal size obeyingthe above conditions with the annulus given by AR
2 ,1, AR2 ,n and everything in
between. (Note that if we had some collection of annuli which bounded smallenergy annuli in a cyclic fashion, we would obtain a connected component of Cwhich had energy less than ǫ, and was not stable.)
We shall now check that all the conditions we require are satisfied by thisnew set of annuli. First, the number is bounded independent of f . The sizeof each annulus is greater than 2R. Our resulting set of annuli also obey thenon-intersection condition 2 because the original set did. We showed abovethat the energy of each new annulus is less than ǫ. Because the complement of
53
our original set of annuli obeyed condition 4 with a bound of 2R(3 + 5Eǫ), by
increasing our conformal bound appropriately and using Lemma 3.16, we canachieve condition 4. The lower bound on the ω-energy of unstable componentsis also satisfied by construction.
All that remains is condition 5. Note first that the complement of our annuliAR,i admits metrics as described in 5. The fact that there exists some c so thatthe metrics we choose on different components differ by a factor of less than con A 6R
10 ,i − A 9R10 ,i follows from conditions 4 and 2, and the bound on topology
as follows: The complement of AR,i is a Riemann surface with bounded genus,a bounded number of boundary components and bounded conformal geometryfrom condition 4. An easy compactness argument tells us that there exists aconstant c′ so that any injective holomorphic map of the unit disk into either Ai
or the complement of AR,i has derivative at 0 bounded by c′. We can use this(remembering that R > 4π > 10 so we can get a disk of unit size centered onany point inside A 6R
10− A 9R
10) to get the ratio of the metric on the complement
of AR,i divided by the metric on Ai is bounded by c′ here.Then, we can use Lemma 3.16 and our conformal bound from condition
4 to get some lower bound r0 > 0 so that the boundary of A 9R10
and A 6R10
is
further than r0 from the boundary of the complement of AR,i. A second easycompactness argument then tells us that if we fix any distance r0 > 0, thereexists some constant cr0 (only depending on the bounds in the above paragraph)so that for every point further than r0 from the boundary there exists an injectiveholomorphic map of the unit disk sending 0 to that point, with derivative at 0greater than cr0 . We then get that the above ratio of metrics above is boundedbelow by
cr0c′
.We must now prove our derivative bound on the complement of A 9R
10. We
shall do this by proving that given any holomorphic injection of the unit diski : D −→ C−⋃AR,i so that i(0) /∈ A 9R
10 ,i, then |d(f i)| is bounded. Then, ourconformal bound from condition 4 will tell us that we have a bound on |df | inthe metric that we have chosen.
We do this in two cases. First, suppose that i(0) is in the complement ofAR
4 ,i for all our old annuli. Then, Lemma 3.16 tells us that the image of i
restricted to the disk of radius e−16π2
R is contained in the complement AR2 ,i for
all our old annuli, as argued above. If the derivative of f i on this disk waslarge enough, we would be able to add another annulus to our old collection inthe manner described above. As this process terminated, |d(f i)| is bounded.Second, suppose that i(0) is contained inside AR
4 ,i for one of our old annuli.
Then Lemma 3.16 tells us that the restriction of i to the disk of radius e−16π2
R
must be contained inside Ai, and therefore have energy less than ǫ. This means
that |d(f i)| ≤ c0e16π2
R , and our derivative is bounded as required.
4 Compactness
Theorem 4.1. Given a basic, complete exploded T fibration B with a civilizedalmost complex structure J and strict taming Ω, the moduli stackMg,n,E(B) ofstable holomorphic curves with a fixed genus g and number of punctures n, and
54
with Ω energy less than E is topologically compact in C∞,δ for any 0 < δ < 1.More generally, if (B −→ G, J,Ω) is a family of such (B, J,Ω), the map
Mg,n,E(B→ G) −→ G is C∞,δ topologically proper.
In particular, this means that given any sequence of the above holomorphiccurves in the fibers over a topologically convergent sequence in G, there existsa subsequence f i which converges to a holomorphic curve f as follows: Thereexists a sequence of families of smooth curves,
(C, ji)fi
−→ B
↓ ↓F −→ G
so that this sequence of families converges in C∞,δ to the smooth family
(C, j)f−→ B
↓ ↓F −→ G
and a sequence of points pi → F so that ⌈pi⌉ → ⌈p⌉, f i is the map given by
the restriction of f i to the fiber over pi, and f is given by the restriction of fto the fiber over p. Of course, the case where we just have a single (B, J, ω) isthe same as the family case when G is a point.
The definition of C∞,δ convergence can be found in section 2.4 starting onpage 18 and the definition of the moduli stack is contained in section 2.8, whichstarts on page 33. The notion of topological convergence is introduced on page10.
The proof of this theorem uses Lemma 3.13 on page 44 and Proposition 3.17on page 49. Together, these allow us to decompose holomorphic curves intopieces with bounded behavior. A standard Arzela-Ascoli type argument gets atype of convergence of these pieces. If we were working in the category of smoothmanifolds, this would be sufficient to prove the compactness theorem becausethe type of convergence involved would have a unique limit, and therefore thelimiting pieces would glue together. The extra problem that must be dealt within this case is that we are not dealing with a type of convergence that has aunique limit, so we must work much harder to show that our resulting ‘limitingfamily’ for each piece glues together to the limiting family F.
Example 4.2.
The following are examples of the types of non uniqueness we have to dealwith:
1. In T11, let z(pi) = 1
it0. Then ⌈pi⌉ → ⌈p⌉ where p → T1
1 is any point sothat z(p) = cta where a > 0. Note that there is no ‘point’ p→ T1
1 so thatz(p) = 0t0. (The author did try modifying the definitions of explodedfibrations so that such limits exist, however if this is tried, the theory ofholomorphic curves becomes much more complicated, and the whole setupis a lot less natural.) This is one reason that non unique limits need to beconsidered.
55
2. Expanding example 1, consider the maps f i : CP 1 −→ CP 1×T11 given by
f i(u) = (u, pi). This sequence of maps does not have a unique limit. Weinstead consider the family
CP 1 × T11
f :=id−−−→ CP 1 × T11
↓T11
Then the sequence of maps f i are given by the restriction of the inverseimage of f to the inverse image of pi. These converge topologically to frestricted to the inverse image of p, where p → T1
1 is any point so that⌈pi⌉ → ⌈p⌉.
3. Expanding example 1 to see different behavior, consider the map π :T22 −→ T1
1 so that π∗z = w1w2. This was studied in example 2.47 onpage 26. This can be considered to be a family of annuli by restrictingto the subset where |w1| < 1 and |w2| < 1. Then π−1(pi) is an annulus1 > |w1| > 1
i, or 1
i< |w2| < 1. These domains converge topologically to
π−1(p) where p = cta for any c ∈ C∗ and a > 0. π−1(cta) has two coordi-nate charts which are subsets of T1
1, with coordinate wi so that |wi| < 1,and ⌊wi⌋ > ta. The transition map between these coordinate charts isgiven by w1 = ctaw−1
2 .
4. Let us expand on example 3 by adding in the information of maps f i fromour domains π−1(pi) to T2. Suppose that
f i(w1) = (ci1w1, ci2w1)
so of course,
f i(w2) = (ci11
iw−1
2 , ci21
iw−1
2 )
If (ci1, ci2) ∈ T2 are generically chosen, we will not have a unique limit even
if we restrict to the bounded domain 1 > |w1| > 12 . To put all our f i into
an individual family, consider the family
T2 × T11
f(c1,c2,w1,w2):=(c1w1,c2w1)−−−−−−−−−−−−−−−−−−−→ T2
↓ id×πT2 × T1
1
Our individual maps f i are the restriction of f to the fiber over (ci1, ci2, p
i).
These converge topologically to the restriction of f to the fiber over anypoint (c1, c2, ct
a) where a > 0. Note that if we restricted to the domainswhere 1 > |w1| > |w2| or 1 > |w2| > |w1|, then we get a T2 worth of validtopological limits for each domain. There is a non trivial requirement thatneeds to be satisfied for these topological limits to be glued together.
Apart from an easy use of an Arzela Ascoli type argument and elliptic boot-strapping, the main hurdle to proving Theorem 4.1 is dealing with the abovetypes of non uniqueness for limits in showing that the different limiting pieces
56
of our holomorphic curves glue together. This is not extremely difficult, but itrequires a lot of notation to keep track of everything.
Note the assumption thatB is basic is mainly for convenience in the followingarguments. In the family case, by restricting to a subset of G which containsthe image of a subsequence, we can also assume that B is basic.
The following lemma will give us good coordinate charts on B or B. Recallthat if B is basic, we use the notation ⌊Bi⌋ for a strata of the tropical part ⌊B⌋,and ⌊Bi⌋ for the polygon which after identifying some strata is equal to theclosure of ⌊Bi⌋ ⊂ ⌊B⌋. A neighborhood of a strata Bi is then equal to an opensubset of ⌈Bi⌉⋊Tn
⌊Bi⌋using the construction of example 2.17 on page 12. This
has a (sometimes defined) action of Tn action corresponding to the (sometimesdefined) action of Tn on Tn
⌊Bi⌋given by coordinate wise multiplication. We will
need this Tn action to move parts of our holomorphic curves around.
Lemma 4.3. Given any basic exploded T fibration B, and a family B −→ G,for each strata Bi ⊂ B, there exists some Ui ⊂ B containing Bi so that
1. The image in the smooth part ⌈Ui⌉ ⊂ ⌈B⌉ is an open neighborhood of⌈Bi⌉ ⊂ ⌈B⌉.
2. If ⌊Bi⌋ is n dimensional, then there is an identification of Ui with an opensubset of ⌈Bi⌉⋊ Tn
⌊Bi⌋using the construction on page 12. We can make
this identification so that the (sometimes defined) free action of Tn givenby considering ⌈Bi⌉⋊ Tn
⌊Bi⌋⊂ ⌈Bi⌉⋊ Tn satisfies the following:
(a) The Tn action preserves fibers of the family B −→ G, in the sensethat if p1 and p2 have the same image in G, then z ∗ p1 and z ∗ p2also have the same image in G.
(b) For any point p −→ Ui, the set of values in Tn for which this actionis defined is nonempty and convex in the sense that it is given by aset of inequalities of the form
|zα| > ctx ∈ RtR
where ctx depends smoothly on p −→ B.
(c) If ⌊Bi⌋ is in the closure of ⌊Bj⌋, then the action of Tn on Ui ∩Uj isequal to the action of a subgroup of the Tm acting on Uj.
Proof:The only point in the above lemma that requires proof is the compatibility
of the actions of Tn with restriction and the family. We can construct thisfirst for the strata ⌊Bi⌋ with the highest dimension and then extend it to lowerdimensional strata.
Being able to do this amounts to constructing coordinate charts identifiedwith open subsets of Rk × Tn
B, where coordinates on the projection of thisto G consist of some sub collection of these coordinates (the normal form forcoordinates on a family), and so that transition functions are of the form
(x, w) 7→ (φ(x), f(x)taw)
57
where φ is a diffeomorphism, and f is a smooth C∗ valued function. Thesetransition functions preserve a Tn action of the form
z ∗ (x, w) = (x, zw)
Because of the assumption that the closure of strata in ⌊B⌋ are simply con-nected, we can first reduce to the case that coordinate charts are in the normalform for a family, and transition functions are of the form
(x, w) 7→ (φ(x), f(x,w)taw)
where f is smooth and (C∗)n valued. There is then no obstruction to modifyingour coordinate charts one by one so that the transition functions no longerhave any dependance on w. (This amounts to replacing the coordinates w
with f(x,w)f(x,0) w. This change is well defined on the intersection with previously
corrected coordinate charts, does not affect our charts being in the normal formfor families, and there is no obstruction to extending it to the rest of a chart.)
We shall now start working towards proving Theorem 4.1. The proof shallrely on Lemma 3.13 on page 44 and Proposition 3.17 on page 49. We shall beusing the notation from Proposition 3.17. First, choose the following:
1. Choose a metric, a taming form ω ∈ Ω, and finite collection of coordinatecharts on B each contained in some Ui from Lemma 4.3 and small enoughto apply Lemma 3.13. For the case of a family, restrict B −→ G to somesmall coordinate chart on G in which the image of some subsequenceconverges, and choose our finite collection of coordinate charts on (the
smaller, renamed,) B satisfying the above.
2. Choose exploded annuli Ain ⊂ Ci satisfying the conditions in Proposition
3.17 with ω-energy bound ǫ small enough and R large enough that eachAi
6R10 ,n
is contained well inside some Ui in the sense that it is of distance
greater than 2 to the boundary of Ui. (The fact that we can achieve thisfollows from Lemma 3.11 on page 42.) Also choose ǫ small enough andR large enough so that each connected smooth component of Ai
6R10 ,n
is
contained in one of the above coordinate charts and can have Lemma 3.13applied to it.
Our taming form ω is a smooth two form on some refinement of B. Asconvergence in a refinement is stronger, we will simply call this refinement B.
Use the notation Cim to indicate connected components of Ci−⋃k A
i8R10 ,k
. We
can choose a subsequence so that the number of such components is the same foreach Ci, Ci
m has topology that is independent of i, and choosing diffeomorphismsidentifying them, Ci
m converges as i → ∞ to some Cm in the sense that themetric from Proposition 3.17 and the complex structure converges to one onCm.
58
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
derivative bounded here
components with ω energy bounded below
AAR:well behaved annulus
ω-energy < ǫ5
Aω-energy > ǫ0
ARAR
2
New low energy annulusDisk with ω-energy > ǫ0New low energy annulus
AR2
AR
A1
A2
A3
A4AR
2 ,[1,3] has low ω-energy
Ci1 = (C1, ji)
(C2, ji)
(C3, ji)Ai
4
Ai5
Ai6
Transition functions bounded
We shall choose two coordinate charts on Ain with coordinates z±n in a subset
of T11 so that each boundary of Ai
n is identified with |z±n | = 1, and the coordinates
are related by z+n z−n = Qi
n ∈ R∗tR+
. (This means that the conformal modulusof Ai
n is − logQin. These Qn will later become coordinates on our family F.)
Choose a subsequence so that the number of annuli is independent of i, andQi
n converges topologically. (In other words, calling any conformal modulusnot a real number infinity, either the conformal modulus of Ai
n converges to afinite number, or converges to infinity. )
We can defineA±
n := ∣
∣z±n∣
∣ ≤ e−7R10 ⊂ T1
1
Ai7R10 ,n
:= (z+n , z−n ) so that z+n z−n = Qi
n ⊂ A+n × A−
n
We shall need to keep track of what An is attached to, so use the notation n±
to define Cin± as the component attached to the end of Ai
n with the bound-ary |z±n | = 1. (Assume, by passing to a subsequence, that n± is well definedindependent of i.)
We can consider transition maps between Cin± and Ai
7R10 ,n
to give transi-
tion maps between Cn± and A±n . The bound on the derivative of transition
maps from Proposition 3.17 on the region Ai9R10 ,n− Ai
6R10 ,n
(and standard el-
liptic bootstrapping to get bounds on higher derivatives on the smaller regionAi
8R10 ,n−Ai
7R10 ,n
) tells us that we can choose a subsequence so that the transition
maps between Cin± and A±
n converge to some smooth transition map betweenCn± and A±
n . We can modify our diffeomorphisms identifying Cim with Cm so
that these transition maps all give exactly the same map. Denote the complexstructure on Cm induced by this identification by ji. (We shall do somethingsimilar for the annuli Ai
n later on.) We can do this so that ji and the metricsgiven by this identification still converge in C∞ to those on Cm.
59
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
derivative bounded here
components with ω energy bounded below
AAR:well behaved annulus
ω-energy < ǫ5
Aω-energy > ǫ0
ARAR
2
New low energy annulusDisk with ω-energy > ǫ0New low energy annulus
AR2
AR
A1
A2
A3
A4AR
2 ,[1,3] has low ω-energy
Ci1 = (C1, ji)
(C2, ji)(C3, ji)
Ai4
Ai5
Ai6
Transition functions bounded
(C1, j)
q1
(C2, j)
q2
(C3, j)
q3A+
4
4+ := 1
A−4
4− := 2
A+5
5+ := 2
A+6
A−6
6+ := 3
6− := 2
We shall now start to construct our family C −→ F. The first step shall beto get some kind of convergence for each piece of our holomorphic curve.
For each Cm choose some point qm ∈ Cm, and consider Qim := f i(qm) ∈ B.
This sequence has a subsequence that converges in ⌈B⌉ to some pointQm −→ B.
Label the strata of B that contains Qm by Bm, and consider the chart Um fromLemma 4.3 containing Bm. As the derivative of f i restricted to Cm is bounded,we can choose a subsequence so that f i(Cm) is contained well inside Um. (Wecan choose a subsequence so that the image of f i(Cm) is contained in the subsetof Um which consists of all points some arbitrary distance from the boundaryof Um.)
As the derivative of f i is uniformly bounded on Ci − ⋃Ai9R10 ,k
, we can use
lemmas 3.6, 3.7, and 3.8 to get bounds on the higher derivatives of f i restrictedto Ci
m. Lemma 3.1 then tells us that if f i(qm) converges topologically to Qm −→B, the geometry around f i(qm) converges to that around Qm, so we can choosea subsequence so that f i restricted to Ci
m converges in some sense to a map
fm,Qm: Cm −→ B so that fm,Qm
(qm) = Qm ⊂ Um
More specifically, remembering that everything is contained inside Um, we canuse our Tn action on Um to say this more precisely. There exists some sequencecim ∈ Tn so that
cim ∗ f i : Cm −→ Um
converges in C∞ to fm,Qm: Cm −→ Um.
Of course, there was a choice involved here. We could also have chosen adifferent topological limit Q′
m of f i(qm), which would give a translate of fm,Qm
by our Tn action. These choices will turn up as parameters on our family.They will need to be ‘compatible’ in the sense that these pieces will need to fittogether.
We now consider the analogous convergence on annular regions. Becausethe ω-energy of f i restricted to our annular regions Ai
n is small, we can applyLemma 3.13 on page 44 to tell us that if the limit of the conformal modulus ofAi
n is infinite, then f i restricted to the smooth parts of Ain converges in some
60
sense (considered in more detail later) to some unique pair of holomorphic maps
f±n,Q : e− 7R
10 >∣
∣z±n∣
∣ > 0 −→ B
compatible with fn±,Qn±
and the transition maps. (Recall that Cn± is the
component attached to A±n .) As we have made the assumption that J is civilized,
we can use the usual removable singularity theorem for finite energy holomorphiccurves to see that these limit maps extend uniquely to smooth maps on e− 7R
10 >|z±n | > tx ⊂ A±
n for some x > 0.
PSfrag replacements
⌊T11⌋ := tR
+
⌈T11⌉ := C
A picture of [T11]
C∗
[C]⌈C⌉⌊C⌋
⌈ExplM⌉ = M⌊ExplM⌋
v1
v2
v3
v4
[C]⌊f(C)⌋ ⊂ ⌊Tn⌋∑
df(vi) = 0⌈M⌉[M]⌊M⌋
Mi := ⌈Bi⌉Ui,j1
Ui,j2
Mj1
Mj2
Mk
ri,j1ri,j2rj1,krj2,k⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋
⌊r1,j⌋⌊r2,j⌋⌊r3,j⌋ = tc
⌊Bj⌋⌊p⌋⌈B⌉⌊B⌋⌈B′⌉⌊B′⌋
derivative bounded here
components with ω energy bounded below
AAR:well behaved annulus
ω-energy < ǫ5
Aω-energy > ǫ0
ARAR
2
New low energy annulusDisk with ω-energy > ǫ0New low energy annulus
AR2
AR
A1
A2
A3
A4AR
2,[1,3] has low ω-energy
Ci1 = (C1, ji)
(C2, ji)(C3, ji)
Ai4
Ai5
Ai6
Transition functions bounded(C1, j)
q1(C2, j)
q2(C3, j)
q3A+
4
4+ := 1A−
4
4− := 2A+
5
5+ := 2A+
6
A−6
6+ := 36− := 2
C4+ := C1
q1 f1,Q1 : C1 −→ U1
q1 7→ Q1 ∈ U1
f+4,Q−−−→ U4
U4
f−
4,Q←−−−
z+4 z−4 = Q4
C2
f1,Q2−−−→ U2
q2 7→ Q2 ∈ U2
C4− := C2
q2
We may assume after passing to a subsequence that f i(Ai6R10 ,n
) are all con-
tained inside a single Un. Recalling our convention that Cn± intersects Ain, note
that this means that Un± intersects Un (and ⌊Un±⌋ ⊂ ⌊Un⌋), so our sequencecin± defined earlier also has an action on Un, and
cin± ∗ f i : Ai7R10 ,n−→ Un
converges in C∞ on compact subsets adjacent to Cn± to f±n,Q .
We now have a type of convergence of the individual pieces we have cut ourholomorphic curve into. We shall now define a model for the exploded structureon our family that is too large, as it ignores the requirement that these piecesmust fit together.
Define Vm ⊂ Um to be the exploded fibration consisting of all points Q′m −→
Um so that
1. There exists some z so that Q′m = z ∗Qm
2. Definingfm,Q′
m:= z ∗ fm,Qm
The image of Cm, fm,Q′m(Cm) is contained well inside Um in the sense
that the distance to the boundary of Um is greater than 1.
3. If Cm is attached to A±n , then defining
f±n,Q′ := z ∗ f±
n,Q
61
The image of the smooth part of A±n , f±
n,Q′(z), 0 < |z| < e−7R10 is also
contained well inside Un.
Note that Vm is a smooth exploded fibration which includes all Q′m which
are topological limits for Qim. For such Q′
m, the map fm,Q′mwill be holomorphic,
but for other Q′m, this may not be the case.
We shall consider the family F as a sub exploded fibration of
∏
m
Vm
∏
n
Qn ⊂∏
m
Um
∏
n
Qn
The Qn above stands for the ‘gluing parameter’ for identifying coordinatesz±n on A±
n viaz+n z
−n = Qn ∈ Qn := |z| < e−2R ⊂ T1
1
We shall consider the following sequence of points
Qi := (f i(q1), fi(q2), . . . , Q
in1, . . . ) ∈
∏
m
Um
∏
n
Qn
where the conformal modulus of Ain is equal to − logQi
n.
Note that there is a transitive (sometimes defined) action of Tk on∏
Vm
∏
Qn
which is the action from Lemma 4.3 on Vm ⊂ Um, and multiplication by somecoordinate of Tk on Qn. Our family will be given by a complete inclusion
F −→∏
m
Vm
∏
n
Qn ⊂∏
m
Um
∏
n
Qn
satisfying the following conditions:
1. The image of F contains some point Q −→ ∏
m Vm
∏
n Qn which is atopological limit of Qi, and the image of F is given by the orbit of Qunder the action of some subgroup of Tk.
2. The distance in any smooth metric from some subsequence of Qi to theimage of F converges to 0.
3. There is no other inclusion satisfying the above conditions which hassmaller dimension than F.
It is clear that such an F must exist. It can be seen from Lemma 4.3, item2 that F is smooth.
Now let us construct C −→ F. We shall have charts on C given by Cm :=Cm × F and An. An has coordinates (z+n , z
−n , Q) where z±n ∈ A±
n and Qn =
z+n z−n . Transition maps between Cm and An are simply given by the transition
maps between Cm and A±n times the identity on the F component. As an
explicit example, if φ is a transition map between Cm and A+n , the corresponding
transition map between Cm and An is given by
(z,Q) 7→(
φ(z),Qn
φ(z), Q
)
62
This describes the exploded fibration C. The map down to F is simply givenby the obvious projection to the second component of each of the above charts.This gives a smooth family of exploded curves. Note that condition 6 fromProposition 3.17 ensures that this is a family of stable curves.
Lemma 4.4. If Q −→ F, and ⌈Qin⌉ → ⌈Qn⌉, then f±
n,Q can be glued by the iden-
tification z+n z−n = Qn. This is automatic if the limit of the conformal modulus
of Ain is finite.
Proof:In the case that the limit of the conformal modulus of Ai
n is finite, limQin =
Qn for anyQ −→ F. If this was not true, we could simply restrict F to the pointsthat satisfy this. This would include any point in F that is the topological limitof Qi, it would also satisfy the other conditions required of F, but have smallerdimension contradicting the minimality of F. When we have this, as we have f i
converging on Cn± and Ain, and transition maps between these also converging,
we can glue the limit.In the case that the limit of the conformal modulus of Ai
n is infinite, we wantto glue f+
n,Q(z+n ) to f−
n,Q(z−n ) over the region where the smooth coordinates
⌈z+n ⌉ = ⌈z−n ⌉ = 0 via the identification z+n z−n = Qn. Define the following
continuous function
φn : F −→ Tl (the group acting on Un)
which detects the failure for f±n to glue for any Q −→ F: If ⌈Qn⌉ = 0, and
z+n z−n = Qn so that ⌈z+⌉ = ⌈z−⌉ = 0, let φ(Q) be the element of Tl so that
φn(Q) ∗ f+n,Q(z
+n ) = f−
n,Q(z−n ). This may only fail to be defined because the left
or right hand sides might not be inside U. Note first that if defined, this doesn’tdepend on our choice of z±n . Also note that this will be defined on some opensubset of F, and there exists some homomorphism from Tk to Tl so that thismap is equivariant with respect to the Tk action on F and the Tl action on Um.We can uniquely extend φn to the rest of F in an equivariant way.
We want to show that φn is identically 1 on F. For each Qi, there exists aclose point Qi inside F so that the distance in any smooth metric between Qi
and Qi converges to 0 as i → ∞. Then φn(Qi) converges to 1, because these
points Qi come from holomorphic curves that are converging on either end ofAi
7R10 ,n
to f±n in a way given by Lemma 3.13. This tells us that the sub exploded
fibration of F given by φ−1n (1) satisfies the conditions 1 and 2 above, so by the
minimality of the dimension of F, it must be all of F. This tells us that if Q−1n
is infinite, then we can glue together f±n without any modifications.
Note that choosing any point Q −→ F so that Q is the topological limit ofQi (such a point must exist by the definition of F), the above lemma allows usto glue together fm,Qm
and f±n,Q to obtain our limiting holomorphic curve f
for Theorem 4.1. If the above lemma held for every point in F, we would haveconstructed our family. As it is, we need to make some gluing choices.
If Qn ∈ C∗ and the limit of the conformal modulus of Ain is infinite, we will
need to make ‘gluing’ choices. We do not use the standard gluing and cuttingmaps as this will not give us strong enough regularity for the resulting family.
63
1. Chose some sequence Qi −→ F so that the distance between Qi and Qi
converges to 0.
We need to take care of the different complex structures obtained by gluingA±
n by Qin and Qi
n. Choose an almost complex structure ji on A+n as
follows: Choose smooth isomorphisms
Φin : A+
n −→ A+n
so that
(a)
Φin
(
z+n)
= z+n for∣
∣z+n∣
∣ ≥ e−8R10
(b)
Φin
(
z+n)
=Qi
n
Qin
z+n for∣
∣z+n∣
∣ ≤ e−R
(c) On the region ⌈z+n ⌉ 6= 0, the sequence of maps Φin converges in C∞
to the identity.
Now define ji on A+n to be the pullback under Φi
n of the standard complexstructure. Using the standard complex structure on A−
n , we then get our
ji defined on An. This is compatible with the ji already defined on Cm, sowe get ji defined on C. Note that ji restricted to the fiber over Qi is thecomplex structure on Ci. From now on, we shall use these new coordinateson Ai
n.
2. We now define the linear gluing map as follows:
(a) Chose some smooth cutoff function
ρ : R∗ −→ [0, 1]
so thatρ(x) = 0 for all x ≥ e−
8R10
ρ(x) = 1 for all x ≤ e−9R10
Extend this toρ : R∗tR −→ [0, 1]
satisfying all the above conditions. (We defined this first on R∗ sothat it was clear what ‘smooth’ meant.)
(b) Given maps φ+, φ− : T11 −→ Ck which vanish at z = 0 define the
gluing map
G(φ+,φ−)(z+, z−) := ρ
(∣
∣z−∣
∣
)
φ+(z+) + ρ(∣
∣z+∣
∣
)
φ−(z−)
Note that if φ+ and φ− are smooth, G(φ+,φ−) : T22 −→ Ck is smooth.
Note also that if φ+ and φ− are small in C∞,δ, then G(φ+,φ−) is smalltoo.
3. We now define a linear ‘cutting’ map: as follows:
64
(a) Choose a smooth cutoff function β : R∗ −→ [0, 1] satisfying thefollowing:
β(x) + β(
x−1)
= 1
β(x) = 1 for all x > eR10
(b) Given a map φ : Ain −→ Ck and Qi
n ∈ C∗, where
Ain := z+z− = Qi
n
we can define the cutting of φ as follows:
φ+(z+) := β
(
|z+|2∣
∣Qin
∣
∣
)
φ(z+)
φ−(z−) := β
(
|z−|2∣
∣Qin
∣
∣
)
φ(z−) := β
(
|z−|2∣
∣Qin
∣
∣
)
φ
(
z+ =Qi
n
z−
)
Note that G(φ+,φ−) restricted to z+z− = Qn is equal to φ.
4. Lemma 3.5 tells us that the image of our annuli of finite conformal moduli,f i(Ai
7R10 ,n
) is contained in some coordinate chart appropriate for Lemma
3.13. (We can choose this coordinate chart to be contained inside Un sothat the Tk action on Un just consists of multiplying coordinate functionsby a constant). In fact, we can choose a subsequence so that either Ai
n
has infinite conformal modulus for all i or f i(Ai7R10 ,n
) is contained in one of
these coordinate charts for all i. In that case, Lemma 3.13 together withthe conditions on our coordinate change for Ai
n above tells us that
f in(z) = (eφ
in,1(z)cin,1z
α1 , . . . , eφin,k(z)cin,kz
αk , . . . , cn,d−k + φin,d−k)
:= eφincinz
α
where φin is exponentially small on the interior of Ai
R,n as required byLemma 3.13. We can regard our local coordinate chart as giving us alocal trivialization of TB compatible with our Tk action. We can extendthis trivialization using our Tk action to an open subset On ⊂ Un whichwill contain the image of any relevant translation of f i
n. Note that On is anopen subset of some refinement of Tk×Rd−2k (with the above coordinates).We shall define our maps as maps to Tk×Rd−2k. Note that in particular, ifQ∞ −→ F is some topological limit of Qi, then f±
n,Q∞ in these coordinatesare given by
f+n,Q∞(z+) := eφ
+n (z+)cn(z
+)α
:= (eφ+n,1(z
+)cn,1(z+)α1 , . . . , eφ
+n,k
(z+)cn,k(z+)αk , . . . , cn,d−k + φ+
n,d−k(z+))
f−n,Q∞(z−) := eφ
−n (z−)cn
(
Q∞n
z−
)α
where φ± are smooth and vanish at z± = 0.
Similarly, if Ain is infinite, then we can consider f i
n to map to On so that
65
f in(z
+) := eφi+n cin(z
+)α
f in(z
−) := eφi−n (z−)cin
(
Qin
z−
)α
5. Use the notation p1
p2for two points pi −→ Vn to mean the element of Tk
so that p1 = p1
p2∗ p2. Define F i
n : An −→ B by
F in(z
+, z−, Q) :=Qn+
Qin+
∗ cin(z+)α
and
Fn(z+, z−, Q) :=
Qn+
Q∞n+
∗ cn(z+)α
(Recall that Qn+ recorded the position of Cn+ which is attached to A+n ).
Note that even though we defined this as a map to Tk × Rd−2k, and ourchart is some refinement of this, these F i
n are still smooth maps to B.(This can be seen if we restrict this map to local coordinate charts.)
Then we can define f i : An −→ B using the cutting and gluing mapsabove by
f i(z+, z−, Q) := eG
(φi+n ,φ
i−n )
(z+,z−)F in(z
+, z−, Q)
We can similarly define f : An −→ B.
Also definef : Cm −→ B
byf(z,Q) = fm,Qm
(z)
and define f i on Cm similarly by translating f i appropriately dependingon the difference between Qi
m and Qm.
This gives well defined smooth maps
f i : C −→ B
f : C −→ B
so that f i is given by the restriction of f i to the fiber of C −→ F overQi −→ F. Note that as the vectorfields φ± are vertical with respect toB −→ G, Gφ+,φ− is too, and we have finally constructed our requiredfamilies.
(C, ji)fi
−→ B (C, j)f−→ B
↓ ↓ ↓ ↓F −→ G F −→ G
66
Lemma 4.5.
f i : (C, ji) −→ B
converges in C∞,δ tof : (C, j) −→ B
Proof:We work in coordinates. First note that f i converges in C∞ restricted to f
on Cm. This also holds for An if the size of Ain stays bounded. Note also that
ji converges in C∞ to j.It remains only to show that f i converges in C∞,δ to f on An. Recall that
An has coordinates (z+, z−, Q) where Q −→ F and Qn = z+z−. Note that the
maps F in : An −→ B defined above converge in C∞ to Fn : An −→ B. We have
that
f i(z+, z−, Q) := eG
(φi+n ,φ
i−n )
(z+,z−)F in(z
+, z−, Q)
Similarly, we write
f(z+, z−, Q) := eG
(φ+n ,φ
−n )
(z+,z−)Fn(z
+, z−, Q)
We can choose a subsequence so that φi±n (z±) converges in C∞ on compact
subsets of 0 < |z±| ≤ e−7R10 to φ±
n (z±). Lemma 3.13 and our cutting construc-
tion above tells us that for any δ < 1, there exists some constant c independentof n so that
∣
∣φi±n (z±)
∣
∣ ≤ c∣
∣z±∣
∣
δ
and the same inequality holds for φ±n and all derivatives (with a different con-
stant c). This implies that φi±n converges to φ±
n in C∞,δ for any δ < 1.A quick calculation then shows that G(φi+
n ,φi−n )(z
+, z−) converges in C∞,δ to
G(φ+n ,φ−
n )(z+, z−). Adding the extra coordinates on which G(φi+
n ,φi−n ) does not
depend does not affect C∞,δ convergence, so we get that f i converges to f inC∞,δ as required.
This completes the proof of Theorem 4.1.
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